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Technical Report SL-93-t6 • - __ September 1993 *
AD-A271 540 US Army Corps iMillKlli ill * of Engineers "■■■ / rk n J Waterways Experiment Station
Two-Dimensionai Finite Element Analysis of Porous Media at Multikilobar Stress Levels
by Stephen A. Akers Structures Laboratory
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Approved For Public ROIMM; Distribution is Unlimitod
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Prepared for Discretionary Research Program U.S. Army Engineer Waterways Experiment Station
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The contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products.
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Technical Report SL-93-16 September 1993 i
Two-Dimensional Finite Element Analysis of Porous Media at Multikilobar Stress Levels by Stephen A. Akers
Structures Laboratory
U.S. Army Corps of Engineers Waterways Experiment Station 3909 Halls Ferry Road Vk*sburg,MS 39180-6199 DTI1-
Final report
Approved tor pubkc
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Prepared for Discretionary Research Program U.S. Army Engineer Waterways Experiment Station 3909 HaHs Ferry Road, Vicksburg, MS 39180-6199
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US Army Corps of Engineers Waterways Experiment Station
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Waterway* Experiment Station C«taloging^n-PuWtcation Date
Akers, Stephen A. Two-dimensional finite element analysts of porous media at multikilobar stress
levels / by Stephen A Akers; prepared for Discretionary Research Program, U.S. Army Engineer Waterways Experiment Station.
105 p.: ■.; 28 cm. — (Technical report ;SL-93-T6) wcaioei Dwnograprucai references. 1. Sols —Tasting. 2. Porous materials—Testing. 3. Matter — Properties ~
Mathematical models. 4. Finite element method. I. U.S. Army Engineer Water- ways Experiment Station. II. Discretionary Research Program (U.S. Army Engineer Waterways Experiment Station) 111. Title. IV. Series: Technical report (U.S. Army Engineer Waterways Experiment Station); SL-93-16. TA7W34no.SL-93-16
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Contents
Preface v
1-Introduction 1
Background 1 Approach 2 Purpose and Scope 3
2-Finite Element Model 4
Introduction 4 Background 4 Finite Element Formulation 6
Effective stresses and strains 6 Finite element equations , 7 Equations for residual forces 11
Constitutive Models 15 E!ement Implemented into JAM IS Summary 16
3-The Cap Model and Its Implementation 17
Introduction 17 Background 17 Loading Functions and Flow Rule 19 Derivation of Incremental Elastic-Plastic Stress-Strain Relations 21
Elastic-Plastic Constitutive Matrix 25 Plastic Hardening Modulus 27 Cap Model Implemented into JAM 28 Numerical Implementation of the Cap Model 33
Introduction 33 Elastic algorithm 34 Failure envelope algorithm 34 Cap algorithm .40 Tensile algorithm 42
Implementation of Cap Model into JAM 43 Verification 44
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lit
IV
Summary 49
4-Equations of State for Air, ,*) Water, and Solids 50 »
Introduction 50 4r Equation of State for Water 50 Air-Water Compressibility 52
Background 52 Boyle's and Henry's laws .53 * Derivation of equations 54
Equation of State for Solids 59 Summary 62
5-Features and Verification of FE Program 63 #
Introduction 63 Additional Features of FE Program 63
Restart feature . 63 Postprocessing 64
Plane and Axisymmetric Verification Problems 64
Consolidation Problems 70 Cryer Problem 73 Summary 75
6-Numerical Simulations 76
Introduction 76 Salem Limestone , 76 Simulations 76
Process 76 Drained limestone simulations 77 Undrained limestone simulations 79
Test Specimen Simulations 84 FE grid 84 Simulation of drained triaxial compression test 84
Simulation of consolidated undrained triaxial compression test 88
Summary 91
7-Summary 92
References 93
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Preface
This investigation was sponsored by the U.S. Army Engineer Waterways Experiment Station (WES) under the Discretionary Research Program.
The work was conducted by Mr. Stephen A. Akers, Geomechanics Division (GD), Structures Laboratory (SL), WES, during the period January 1991 through March 1993, under the general direction of Dr. J. G. Jackson, Jr., Chief, GD, and Mr. J. Q. Ehrgott, Chief, Operations Group, GD. Mr. Bryant Mather is Director, SL.
At the time of publication of this report, Dr. Robert W. Whalin was the Director of WES. Commander was COL Bruce K. Howard, EN.
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1 Introduction
Owpttf 1 Introduction
9) 9
Background
The personnel of the Geomechanics Division, Structures Laboratory, U.S. • Army Engineer Waterways Experiment Station (WES) are responsible for research and development in the general field of soil and rock dynamics. Our primary interest is in the response of earth and earth-structure systems subjected to intense transient (blast-type) loadings. An analysis of these systems is typically conducted in three different phases. First, laboratory tests 0 are conducted on the geologic materials of interest in order to develop a data base of composition and mechanical properties; then, based upon this data base, a set of recommended material properties is developed for the consti- tutive modelers. In the second phase, the modelers fit one or more consti- tutive models to the recommended material properties. Last, finite element (FE) or finite difference codes are used by the ground shock calculators to • • analyze the responses of these systems.
The Geomechanics Division is frequently asked to conduct mechanical property investigations. In performing these investigations, we have tested and characterized many types of materials. These materials generally fall into + the following groups: moist and fully saturated cohesionless soils, desert alluviums, natural and remolded clays, clay shales, soil and rock "matching" grouts, and a variety of competent rocks. As basing and attack scenarios of the Department of Defense become more elaborate, and as the analysis tech- niques of modelers and ground shock calculators become more refined, greater demands are placed on the engineer who is asked to perform and • analyze the laboratory mechanical property tests in order to provide recom- mended material properties. Modelers and calculators are now requesting total-stress mechanical p. ^K..V data at stress levels of several kilobars. Complicated stress- and strain-path tests are frequently included in their lists of desired material response tests. Greater emphasis in effective-stress 0 material properties is now evident in the ground shock community.
Due to the unconventional nature of many of the requested tests, the engineer performing and analyzing the tests is sometimes uncertain about existing laboratory equipment, i.e.. whether it restricts the types of tests that can be conducted. The engineer may question the measured laboratory •
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responses; are they theoretically realistic? An engineer may have specific questions such as: (a) what effect will small amounts of air-filled porosity have on material properties, (b) what loading rates are appropriate for con- ducting truly drained tests or undrained tests with meaningful pore pressure measurements, and (c) how does one calculate effective stresses at kilobar stress levels for rock-like materials? In some situations, an engineer respon- sible for recommending material properties may only have low pressure (less than a kilobar) total-stress and effective-stress data from which to extrapolate multikilobar material responses.
An engineer would have a tremendous advantage if a numerical tool were available with which to verify laboratory test results or to predict unavailable laboratory test data. The appropriate numerical tool should give an engineer the capability to calculate both total- and effective-stress material responses. This numerical tool would:
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1. calculate strains, total and effective stresses, and pore fluid pressures for fully- and partially-saturated porous media,
2. calculate the time dependent flow of pore fluids in porous media, 3. model nonlinear irreversible stress-strain behavior, including coupled
shear-induced volume change, and 4 simulate the effect of nonlinear pore fluid compressibility and the
contribution of the compressibility of the grain solids for stresses up to several kilobars.
The FE code JAM incorporates all of the above features. The code simu- lates quasi-static, axisymmetric, laboratory mechanical property tests, i.e., the laboratory tests are analyzed as boundary value problems. Features 1 aid 2 were incorporated into the code using modified formulations of Biot's coupled theory as advanced by investigators such as Zienkiewicz (1985a) and Lewis and Schrefler (1987). An elastic-plastic strain-hardening cap model calculates the time-independent skeletal responses of the porous solids. This enables the code to model nonlinear irreversible stress-strain behavior and shear-induced volume changes. In order to accurately model the total- and effective-stress responses of multikilobar laboratory tests, fluid and solid compressibilities were incorporated into the code. Following the concept used by Chang and Duncan (1983), partially-saturated materials were simulated with a "homoge- nized" compressible pore fluid.
• •
Approach
To develop the FE code, four major tasks were completed. They were:
1) a cap model was incorporated into an existing two-dimensional finite element code PLAST and numerical experiment were conducted to verify its implementation;
CtepMf 1 Introduction
2) a modified version of Biot's coupled theory was implemented into the code produced in step 1 above;
3) a data base of drained and undrained mechanical properties was ob- tained for a limestone with a porosity of 13.5%; it included mechani- cal properties from tests conducted at stress levels of up to 6 kilobars and recommended properties;
4) using the recommended properties from step 3, a cap model was fit to the drained properties; the undrained stress-strain and pore pressure responses of the material were calculated with the FE code and then compared to the undrained material responses.
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Purpose and Scope
The purpose of this report is to document the features and algorithms implemented into the FE code JAM. Chapter 2 describes the FE model im- plemented into JAM and briefly documents the constitutive models available in the code. The essential features of the cap model are reviewed and the steps required to implement the cap model into the FE code JAM are summarized in Chapter 3. The equations of state for air, water, and grain solids are docu- mented in Chapter 4, and the equations for compressibility of an air-water mixture are developed. Chapter S describes features in the FE program not introduced in earlier chapters and presents solutions from several verification problems as proof that the program works correctly. Numerical simulations of limestone behavior under drained and undrained boundary conditions are presented in Chapter 6. The final chapter summarizes the results of this research effort.
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ChtpMt 1 introduction
2 Finite Element Model
Introduction
This chapter describes the FE model implemented into JAM and briefly documents the constitutive models available in the code. The work of Biot and other investigators is described, followed by a discussion of the equations implemented by Lewis and Schrefier and modifications that must be made to those equations. In addition, the equations are derived for the residuals. Finally, the five constitutive models available in JAM for modelling skeletal behavior are described.
Background
In 1941, Biot published his three-dimensional theory of consolidation for static loading. In his theory, Biot coupled the solution of the equations of pore fluid diffusion with the equations of deformation for the porous solids. He was thus able to calculate time-dependent displacements, strains, pore fluid pressures, and effective stresses. Biot made the following assumptions in his formulation: (1) the material was isotropic, (2) the material was linear elastic, (3) small strains were applicable, (4) the pore water was incompressible, (5) the pore water could contain air bubbles, and (6) flow of the pore water obey- ed Darcy's Law. In subsequent papers. Biot extended his theory to include anisotropic materials, viscoelastic materials, and dynamic processes (Biot 1955; 1962).
With the rapid development of digital computers and advances in numerical techniques such as the finite element method, many investigators expanded Biot's theory in attempts to model more realistic and more complex problems. Sandhu and Wilson (1969) were the first to use finite dement techniques with Biot's original formulation to solve initial boundary value problems They applied variational principles to the field equations of fluid flow in a fully saturated porous elastic continuum, and then used the finite element method to numerically solve the resulting coupled equations.
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Chapter 2 Fwut« Etcmani Mo*M
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Ghaboussi and Wilson (1972, 1973) developed a variational formulation of Biot's dynamic field equations for saturated porous elastic solids. Their finite element formulation allowed for the compressibilities of both the fluid and solid phases. Their methods were applicable to dynamic soil-structure inter- ^ action and wave propagation problems in saturated geologic media.
Zienkiewicz and his colleagues have written extensively about their use of modified versions of Biot's formulation to solve consolidation, liquefaction, and wave propagation problems in fluid saturated porous materials (Simon et al. 1986a and 1986b; Zienkiewicz 1985a; Zienkiewicz et al. 1980; Zienkiewicz and Shiomi 1984). They have incorporated several different non- linear constitutive models into their numerical codes. For example, Lewis et al. (1976), used a hyperbolic constitutive relationship to model the skeletal response of the solids. They incorporated fluid and solid compressibilities, creep, and void -atio dependent permeability into their code. Zienkiewicz and his colleagues have also demonstrated a capability to solve dynamic problems such as ground motions due to earthquakes. For example, Zienkiewicz et al. (1982), modified the Critical State model to include a Coulomb-type failure surface and incorporated a "cumulative densification" feature into the consti- tutive model that allowed pore fluid pressures to increase with increasing numbers of load-unload cycles. They then demonstrated the utility of their approach when they used their code to approximate the earthquake induced displacements and pore pressures within the Lower San Fernando Dam.
Other investigators have also expanded Biot's formulation with nonlinear constitutive models. Oka et al. (1986) developed an elasto-viscoplastic consti- tutive model for clay and used Biot's theory to study the two-dimensional con- solidation response of sensitive and aged clay deposits. They demonstrated the capability of their code to simulate the consolidation response of clay deposits during the construction phase of embankments. Chang and Duncan (1983) used a modified Cam Clay constitutive model in their analyses of earth structures constructed of compacted, partially saturated clay soils. They also applied the concept of a "homogenized pore fluid" to Biot's formulation in order to model partially saturated clay soils. With this implementation, they were able to treat partially saturated soils as two-phase materials, i.e., solids and compressible pore fluid, instead of using a more theoretically rigorous approach involving three-phase materials.
Lewis and Schrefler (1987) extended Biot's formulation to include the governing equations for single phase, multiphase, and saturated-unsarurated flow in a deforming porous solid. They discussed finite element procedures for both the space and time discretization aspects of consolidation problems. They also presented linear elastic and nonlinear constitutive relationships; the nonlinear models included the hyperbolic model and incremental elastic-plastic models such as the Critical State models.
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5
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6
Finite Element Formulation
Effective stresses and strains
For a nonlinear material not susceptible to creep strains, a general stress- strain relation can be written as
daij' ' Dijki\d*kl ~ d4if 2.1
Dijkid*ki'
where ai;' is the matrix of effective stresses, Dtjkl is the tangential stiffness
matrix or constitutive matrix, dtkl is the matrix of total strains, dtskl is a matrix of strains due to the compression of the grains by the pore fluid and
dekl' is a matrix of effective strains. The matrix dekt is evaluated as:
where Kg is the bulk modulus of the grains, r is the pore fluid pressure, 6U is the Kronecker delta defined by
«■■-I" * if is"J v \ = 0 if i #j
and an engineering mechanics sign convention is used in which compression is negative.
The purpose of the term dt kl is illustrated in the following example. If a porous specimen surrounded by a pervious membrane was placed into a pres- sure vessel and a pressure of several hundred megapascals was applied, a volume decrease would be measured in the specimen due to the compression of the grains However, since the total strain dtkl is equal to the volume
strain due to grain compressibility, i.e.. dtkl - dtgkl, the effective strains and therefore the effective stresses within the specimen are zero. With the term
dt Kkl included in the general stress-strain relation, effective stresses can
simply be calculated as:
do,/ - dotJ * *6); 2.3
Under drained boundary conditions, the total and effective strains are equal, and the total and effective stresses are equal
Chapter 2 Ftfvw El*m*m Mod«!
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In Equation 2.3, no factor need be applied to the pore pressure term to account for grain compressibility, which was a method proposed by Skempton (1960). After appropriate manipulation, the above equations will yield Skempton's equation
Ap' * Ap 1 - AT 2.4
where p is pressure, AT and K. are the bulk modulus of the skeleton and grain solids, respectively.
Finite element equations
The general equations developed from the spacial discretization of the equilibrium and continuity equations have been documented by a large number of investigators (Lewis and Schrefler 1987; Zienkiewicz 1985a). Lewis and Schrefler developed the following equations:
[K)u - [L)r - R • 0 2.5
and
\H)r~[S)f -[L)Tu - Q*0 2.6
where [K\ is the tangent stiffness matrix of the solid phase.
K - | BTDTBdQ 2.7
[L] is the coupling matrix between the solid and fluid phases.
- f BTm, I - I B'mNdQ - f BTD. 'a 'Q
3K. NdQ 2.8
[W] is the permeability matrix of the porous skeleton.
Tk H - f (VW)1 1 %'Ndü 2.9
(5) ts the compressibility matrix.
S • [ NTs NJQ 2.10
Chaptw 2 Fmitc EMmcni Mod»)
in which the scalar s is evaluated as
. Ill ♦ ± - __L_ m^D Kt Kf
R is the external force vector,
- [ NTdbdQ * [ NTdt
Q is the vector of boundary flows,
5 = i_Z + JL- I mlDTm 2.11 Kg Kf OKg)
2
R - f NTdbdQ * [ NTdtdT 2n
r * Vooun 2.13
8
ß= [Nr9<xT + [ (VN)T 1 VpghdQ
and the superimposed dot indicates a time derivative. In the above equations, (it is the vector of displacement increments, * is the vector of pore pressure increments, D T is the elastic-plastic constitutive matrix, m is the matrix equivalent of the Kronecker delta, B is the strain-displacement matrix, N is the matrix of displacement shape functions, N is the matrix of pore pressure shape functions, b is a vector of body forces, if is a vector of surface trac- tions, k is the absolute permeability matrix of the material, p is the dynamic viscosity of the pore fluid, K. and K, are the bulk modulus of the grain solids and pore fluid, respectively, 4» is the porosity of the material, q is the vector of applied fluid flux, and p, g and h are fluid density, gravity, and elevation head, respectively.
Lewis and Schrefler (1987) describe in detail the components that contrib- ute to the rate of fluid accumulation. However, one of the terms in their formulation is misleading if not incorrect. The following discussion expands on their analysis and then shows why their formulation requires modification. For the material and conditions of interest, Lewis and Schrefler specify that four volumetric strain components must be evaluated; they are the volume
strain of the porous matrix t ™, the volume strains of the grain solids € f and
the pore fluid < /, and a component of solid volume strain < *' due to the applied effective stresses. To simplify the discussion of fluid accumulation, consider the following test conditions. A fully saturated porous material is contained in a sample chamber, which has frictionless sides, and is loaded by a frictionless piston with area A. A flow meter attached to the piston indicates the direction of fluid flow into (positive flow) or out of (negative flow) the sample. The top surface of the piston is loaded by a chamber pressure P, and a second fluid pressure P2 is applied through the flow meter to the pore fluid within the sample. The fluid pressures P| and P2 control the total and effec- tive stresses within the sample. Recall that for a porous material with a
Chapter 2 Finn« Etoflwm Modal
dt\ Ml -n)d(s = (1-/.) Zp. 2.16 Kl
where rfP has been replaced by the pore fluid pressure dr.
In the same manner, the volume change within the unit volume due to the compression of the pore fluid is written as:
d({- -nt. 2.17
where K, is the bulk modulus of the pore fluid.
Lewis and Schrefler also evaluate the component of volume strain due to the compression of the grains caused by the increase in effective stress. A similar analysis was developed by Bishop (1973). Refer again to the test configuration and the example in which P, was increased and P2 held constant For a statistically random distribution of pore space within the
Chapter 2 Ftnrtt Element Mod«!
*7
* ! volume of unity, the volume of the voids is n, and the volume of the solids is 1-n.
If Pj is increased and P2 held constant, the effective stress in the specimen ID will increase, the material will compress, and pore fluid will flow out of the • specimen. Since the specimen is fully saturated with fluid, the rate of change in fluid accumulation is equal to the volumetric strain of the porous matrix * and may be written as:
fC _ ^kk _ ^U s 2.14 » dt ~ dt dt ij
If Pj and ?i are increased at the same rate, the effective stress within the specimen will not change. However, both the grain solids and the pore fluid will compress due to the change in pore fluid pressure. For these conditions, * fluid will flow into the specimen. Let us evaluate the solid and fluid compo- nents separately. The volui .*. strain in the solids due to an applied pressure is expressed as:
dts - d-l - U* 2.15 . ° V, Kg •
where K is the bulk modulus of the grains and Vs is the volume of the sol- ids. From this equation, we can express the volume change within the unit volume due to the compression of the grains as:
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•
9
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specimen, the area of solids (As) on any plane through the specimen will be
A,-(1-nM 2.18
where n is the porosity of the specimen as previously defined. If datj' is the
average normal effective stress on any surface, then datj■,' I (1 - n) is the average normal stress in the solids. Using the concept implied by Equa- tion 2. IS, the pressure applied to the solids can be expressed as:
dP • da:
6t; = ~d0kk' 3(1-n) ",J3(1-n)
which when substituted into Equation 2.16 gives:
de.
2.19
dV dV d°kk' 2.20 (1-«)V 3Kg(\-n)
The component of volume strain due to the applied effective stresses for the unit volume is then:
de ge _ dV da kk
3*. 2.21
The components of the volume strain are now:
de kk de m
dt
del dei 2.22 dt dt dt dt dt
and the expressions for each component when substituted gives:
de total kk de
matrix kk 1 da kk
dt dt 3*, dt 1 n n + _
dr dt
2.23
The term in dispute is the component of volume strain due to the applied effective stresses. Grain compression due to changes in effective pressure is already incorporated into all skeletal constitutive models by default. In an analysis of drained test data, the skeletal strains are not decoupled from the grain strains, and the sum of the two is always measured by strain gauges or deformeters Therefore, both components are included when a constitutive model is fit to drained hydrostatic loading data. A similar argument can be made by examining Equation 2.23. During a drained test, Equation 2.2? would reduce to the following
*Uk dt
^6 dt "
1 do kk IK. dt
2.24
10 Chapter 2 Finite Elamant Model
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Clearly, this is in error; the only strains included in the above expression should be the skeletal strains. Therefore, one should only include grain com- pression due to changing pore pressures in the final formulation. The neces- sary modifications were made to Equations 2.8 and 2.11, which are rewritten below
- | BTm NdQ 2.25
and
, lz± + ± 2.26
Equations for residual forces
Although numerous papers pertaining to the FE equations governing pore fluid flow in a deforming porous solid are available, none outline the equa- tions required to calculate the residual forces. In this section, these equations are developed for a nonlinear incremental finite element program that employs a modified Newton-Raphson solution scheme.
The time integration of Equations 2.5 and 2.6 is performed using the fol- lowing approximation:
k' xdr-aAf'^x Ml -a)Atly 2.27
for 0 £ a <, 1. From Equation 2.27, the following are developed:
'♦Ar t. l*aM, X ' X
At 2.28
and
X * (1-«) X ♦ * X 2.29
Chapter 2 Finn« Element Model 11
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Table 2.1 gives the most common difference schemes adopted by the selec- tion of a given value of a. Equation 2.5 may be written for a given time
Table 2.1. Time Integration Parameters
Value cr Difference Scham« Stability
0 Forward or Euler Conditionally
Vi Crank-Nicolson Unconditionally
% Galerkin Unconditionally
1 Backward Conditionally
t+aAt as:
r + aAfrKi f + aAfj% _ l»ail|M t + aAt- _ t + aAlp 2.30
m
Equations 2.28 and 2.29 are introduced into Equation 2.30 to produce the following
t+&t [K] ""u - 'u Ar
t+M [L] t+iu.
M 2.31
'^'R-'R A/
Multiplying by Ar and collecting terms, one obtains
= ""R-'R .
2.32
In general. Equation 2.32 represents nonlinear behavior. The relationship may be linearized with the following expressions:
t*touU) m /.A/MU l) 4 6uU) 2.33
'M^iD = l'Alfd-ll + 5T(i) 2.34
where i represents the current iteration, and the initial conditions are
<<*u(0) « '„ and "&J0) „ »T This linearization can be used as the first step in a Newton-Raphson iteration (Bathe 1982) If Equations 2.33
12 Chapter 2 Finite E lament Model
Ji
and 2.34 are substituted into Equation 2.32 and the terms for iterations (i-i) and (0) are brought to the right hand side of the equation, then one obtains:
'+A,[tf]««(,) - '+A'[I]«T(0 = ,+AtR - 'R
- '+A'[jf]{'+^tt('-1) - ' + A/M(0)| 2.35
+ fA/j£jjr+A/T(i-l) _ z+A/^OJJ
Recognizing that
tß _ '♦A'rjfi r + ArM(0) _ '+A/r£i t*Atv(0) 2.36
Equation 2.35 may be written in terms of the incremental or accumulative stresses and pore pressures as
2.37
where '♦*/?<'-» = ljr '^'^dVand
/♦A/£.(|-l) _ l + Alr£i(/-l)f*A» (l-l)
Equation 2.37 is one of the two equations required to solve for pore fluid flow in a deforming nonlinear porous solid. The second equation is developed from Equation 2.6 and is written at time t+aAt as:
TU] t*a&IT - f Cl f + orA/^. _ .rJ t*a&ly = t*a&tQ 2.38
Equations 2.28 and 2.29 are introduced into Equation 2.38 to produce
[H]{(\-a)'w *a '♦**} - [S] t*At I T - T
\LV
At
11 Al I
At
2.39
(l-a)'Q * a't&tQ
m
• •
Chap!» 2 Fmrtt El*m«ni Modal 13
.• . A.
14
Collecting terms and multiplying by Ar one obtains:
{-[5] + aAr[ff]} '♦*» - [Lf{ ,+A'«-'«}
+ {[5] + (l-a)Af [*/]}'* = P
where P = Af{(l -a) 'Q + a /+A,ß}. If the following equations
S = -[S] + aAt[H] 2Al
and /7 = [5] *(l-a)Ar[Ä] =Ar[/7] -S 2 42
are substituted into Equation 2.40 for simplification, then one obtains
S ,+A/x - [L]T '**« = P-H'r - [L? 'u . 243
Equation 2.43 must now be formulated for general nonlinear behavior using the same linearization process applied to Equation 2.32, i.e., Equa- tions 2.33 and 2.34 must be substituted. This operation produces
_ / + A/p /»Ar^ f*AifT(0) _ /»A/r^ir »*Artt(0) _ ..
Collecting the bu and 6«- terms on the left hand side of the equation, one gets
'*^S6ir{i) - '*A/[Llr^tt(', * ,tAJp- '♦Af£i*A/f.(Q) 2.45
_ r*Af£ fAf^i-l) 4 '**{£, )r{ »♦Ala(«'D - '«A»,,«»}
Equations 2.41 and 2.42 can be used to eliminate the terms H and S from the right hand side of Equation 2.45 to produce
"*'S6irU) - '*A,[L)T6uU) - "*P 2.46
1*1
2.40 (*)
*J
- A/ ''toff l*Mv(0) _ fAi£<«-l) + t'&iyii 1)
where "^G(l l) • < - '^'|S](,,) ♦ aA/ "*||/]<'-'>} "^A*(I 1). »♦A/^o-n B f«Aij£ir((-i) ci/^jjd-D
Ci/ilf(i-li , /«A/yd!) „ l<i/fl0|
f«A/Aw<«I) . fA/^U-l) . I'il^lO)
Chapter 2 Firwe «lament Modal
• •
• «.
Ar]
Equations 2.37 and 2.46 are written in matrix form for increment f+Ar as:
K - L
- LT - S * aAtH
«««>
«*<''>
R -Fu-n + CO-D
•
•SO
2.47 This is the system of equations that must be solved to calculate displacements and pore fluid pressures in a deforming porous solid.
Constitutive Models
Four of the five constitutive models available in the FE code JAM were deve oped by Owen and Hinton (1980). These models were implemented by Owen and Hinton in the FE code PLAST, which was the original FE code on which JAM was built. Each of the four models were implemented as elastic- plastic models with linear strain hardening, and each has a different yield criteria. The four models include the Tresca and von Mises criteria, which are suitable for metal plasticity, and the Coulomb and Drucker-Prager criteria, which are more suitable for the simulation of soil, rock, and concrete.
The yield stresses in both the Tresca and von Mises criteria are indepen- dent of pressure, which makes these models unsuitable for simulating the pres- sure dependent material behavior exhibited by soil, rock and concrete. In contrast, yield stresses in the Coulomb and Drucker-Prager criteria are pres- sure dependent. For more information on these models, the reader should refer to Chapter 7 in Owen and Hinton (1980).
An elastic-plastic strain-hardening cap model was implemented into JAM to calculate the time-independent skeletal response of the porous solids. The cap model enables the FE code to model nonlinear irreversible stress-strain behav- ior and shear-induced volume changes. Chapter 3 contains extensive docu- mentation on the cap model.
Element Implemented into JAM
A new element was implemented into JAM to calculate both the displace- ment and pore fluid pressures. JAM uses an eight-node isoparametric element with 16 displacement and four pore fluid degrees of freedom. Similar ele- ments were used by Lewis and Schrefler (1987), Simon et at (1986a, !986b). Zienkiewicz (1985a), Zienkiewicz et at (1980). and Zienkiewicr and Shiomi (1984) Four Gauss integration points are utilized in each element
Chapter 2 Ftrwte Ettrnem Model 15
■ ■-
*J
Summary
In this chapter, the work of Biot and other investigators was briefly de- scribed, followed by a discussion of the modified Biot equations implemented by Lewis and Schrefler and modifications that were made to those equations. In addition, equations were derived for the residual forces. Finally, the five constitutive models available in JAM and the element implemented into JAM were described.
5/
• •
16 Ch»ptt' 2 Ftrate Etomcni Model
• 9,
3 The Cap Model and Its Implementation
Introduction
The "cap model falls within the framework of the classical incremental theory of work-han'ening plasticity for materials that have time- and temper- ature-independent properties" (Chen and Baladi 198S). When modelling geo- logic maten?!i subjected to stresses ranging from one to several hundred megapascals, the cap model has several desirable features. Of primary importance is its ability to model volumetric hysteresis through the use of a strain-hardening yield surface or cap.
In this chapter, the essential features of the cap model are reviewed, and the steps required to implement the cap model into the finite element code JAM are summarized. After a brief evaluation of the loading function and flow rule, the incremental elastic-plastic stress-strain relations are outlined. In addition, the cap model implemented into JAM is described, and the equations are developed for the elastic-plastic constitutive matrix and the plastic hardening modulus. Finally, the numerical implementation of the cap model itself is described. The reader should note that an engineering mechanics sign convention is used in which compression is negative.
Background
The cap model has been used by researchers in the ground shock commu- nity for approximately 20 years to simulate the responses of a wide variety of geologic materials. It is "predicated on the fact that the volumetric hysteresis exhibited by many geologic materials can also be described by a plasticity model, if the model is based on a hardening yield surface which includes con- ditions of hydrostatic stress* (Sandler et al. 1976) The model was first described in the open literature by DiMaggio and Sandier (1971). The FORTRAN source code for the model was published by Sandier and Rubin (1979)
Chawtr 3 Th* C»p Mod«) and ft* lmp*»m#oi*tK>n
• •
17
3. ..A.
18
As stated above, the cap model has several desirable features. Of primary importance is its ability to model volumetric hysteresis through the use of a strain-hardening yield surface or cap. The cap model may also be formulated with a nonlinear failure surface, with linear or nonlinear elastic moduli, or as *} a function of the third stress invariant. With the appropriate selection of 9
material parameters, it can be used as a linear elastic or linear elastic-perfectly plastic material model. Sandier and Rubin (1979) demonstrated notable fore- * sight with their use of function statements within the model, which allow sub- stantial changes to be made to the cap model's potential functions with little programming effort.
Modifications and expansions of the original model were made by several researchers. Effective-stress versions of the cap model are reported in Baladi (1979), Baladi and Akers (1981), and Baladi and Rohani (1977, 1978, 1979). A transverse-isotropic cap was developed by Baladi (1978) and an elastic-viscoplastic cap model by Baladi and Rohani (1982). Rubin and * Sandier (1977) developed a high-pressure cap model for ground shock calcu- lations due to subsurface explosions. Baladi (1986) developed a "complex" strain-dependent cap model, which required 39 model parameters, for ground shock calculations of a dry cemented sand. In addition, several versions of the cap model are described in the text by Chen and Baladi (1985).
In formulating the cap model, DiMaggio and Sandier (1971) complied with the constraints imposed by Drucker's stability postulate. Drucker's stability postulate is sufficient, although not necessary, to satisfy all thermodynamic and continuity requirements for continuum models (Sandler et al. 1976). Satisfying Drucker's stability postulate insures uniqueness, continuity, and • # stability of a solution and provides a mathematical problem that is properly posed. Rubin and Sandier (1977) state that "...the numerical solution to a properly posed problem can proceed without the fear that the results will be strongly dependent on errors of approximation of initial and boundary condi- tions, round off error, etc." Drucker (1951) defines a work-hardening mate- rial as one that remains in equilibrium under an added set of stresses applied * by an external agency. It also means that "(a) positive work is done by the externr.1 agency during the application of the added set of stresses and (b) the net work performed by the external agency ever the cycle of application and removal is positive if plastic deformation has occurred in the cycle" (Drucker 1951). Drucker (1950) states these two conditions in a mathematical • format as
daudttJ>0 and do tJdtpt] J> 0
The first statement constrains a model such that strain-softening may not occur The second statement implies (a) the loading function or yield surface must be convex and (b) the plastic strain increment vector must be normal to the yield surface, which means that an associated flow rule must be used. Ihese are the constraints imposed by Drucker's stability postulate
Chapter 3 The Cap Model and 111 Implementation
I • -•-. L
mmmm
Loading Functions and Flow Rule
Drucker's criteria for stability permits considerable flexibility in the func- tional forms of the loading function /. Since Drucker's stability postulate requires the yield surface and plastic potential surface to coincide, the loading function / implicitly represents both the yield and potential surfaces. For a perfectly plastic material, a general form of the loading function may be writ- ten as
/(*«?) -0 3.1
and as
f(ov,K)>0 3.2
for a strain- or work-hardening material, where K is a hardening parameter that acts as an "..internal state variable that measures hardening as a function of the history of plastic volumetric strain" (Sandier and Rubin 1979). For isotropic materials, the loading function may be expressed in terms of stress invariants, e.g.,
/(/,./^".«)-0 33
where J{ = akk = the trace of the stress tensor and
J2p = ttSySij = the second invariant of the deviatoric stress tensor. This is the form of the loading function used in most versions of the cap model. The loading function is assumed to be isotropic and is comprised of two surfaces, an ultimate failure envelope and a strain-hardening surface or cap. The failure envelope, which is fixed in space and symmetric about the hydrostatic axis, limits the maximum shear stresses in the material and is ex- pressed as
The cap, which moves as plastic deformations occur, is represented as
The hardening parameter * is generally taken to be a function of the plastic volume strain (Chen and Baladi 1985)
Cft tpwf 3 Trw Cap Mod*! and I» Imptamantation 19
•1
n (fP \ 3.6
Equation 3.6 allows the cap to expand and contract. By allowing the cap to contract, one can limit the amount of dilation that a material may develop when its stress path moves along the failure envelope h. This form of the hardening parameter is typically used for soil-like materials that do not exhibit significant dilation during failure. For rock-like materials, the hardening parameter may be written as
In this form, the cap is only permitted to expand, thus allowing a material to dilate while its stress path moves along the failure envelope, i.e., when h « 0. Both Equations 3.6 and 3.7 produce hysteresis during an imposed hydrostatic load-unload cycle (Baladi and Akers 1981).
The plastic loading criteria for the loading function / are given by
d°ij
0 loading 3 g 0 neutral loading 0 unloading
(Baladi and Akers 1981). These criteria imply that during loading from a point on a given yield surface a stress increment tensor do^ (when viewed as a vector) will point outward (Rohani 1977). Plastic strains will only occur under this condition. During unloading, the stress vector points inward, and the material will behave elastically. Neutral loading occurs when the stress vector is tangent to the yield surface. During neutral loading, no plastic strains are produced in the case of a work-hardening material (Rohani 1977). This is referred to as the "continuity condition', and its satisfaction leads to the coincidence of elastic and plastic constitutive equations (Chen and Baladi 1985)
Drucker (1951) has shown that the plastic strain increment tensor for a work-hardening material may be written as
*?.
a/ if f » n »ivi d/ dX -£L if / - 0 and 1L. do.. > 0 *°v *a,j J 3.9
0 if / < 0. or / - 0 and M- datJ s 0 v
20
which is identical to the expression used for elastic-perfectly plastic materials The term d X is a positive factor of proportionality that is nonzero only when
Chapter 3 Tha Cap Model «rid H* tmptenwntatioA
plastic deformations occur (Baladi and Akers 1981). For the cap model, the loading function / may take the form of either Equation 3.4 or 3.5.
Derivation of Incremental Elastic-Plastic Stress- Strain Relations
The basic premise in the formulation of the cap model and all elastic-plas- tic constitutive models is that certain materials are capable of undergoing small plastic (permanent) strains as well as small elastic (recoverable) strains during each loading increment (Baladi and Akers 1981). This may be ex- pressed mathematically as
dt . dt' * dt? 31°
where dt,. = components of the total strain increment tensor,
dt*t. = components of the elastic strain increment tensor, and
dt1-: = components of the plastic strain increment tensor.
This equation simple states that the total strain increment is equal to the sum of the elastic and plastic strain increments. In its most general form, the elastic strain increment tensor may be expressed as
where Cljkl(amn) « the material response function, which may be a function of stress. For isotropic materials, the elastic strain increment tensor may be expressed as
. dJt ds„ i ,-» dt - _ 6 ■ ♦ —1 3 12
» 9 A 1 2G
where s - a - (/,/3)6 ■ the deviatoric stress tensor,
6tJ « the Kronecker delta, and A* and C are the elastic bulk and shear moduli, respectively.
The elastic bulk and shear moduli may be constants or functions of stress or strain invariants, e.g..
G * 0{Jx,JlDJlD)
where Jyt> » r/»sIJsjkskl ■ the third invariant of the deviatoric stress tensor.
21 ChapMw 3 Th« Cap Moöai and H» Imetainamaiion
Chen and Baladi (1985) discuss the thermodynamic restrictions to the pos- sible forms of the above equations. Permissible functional forms of K and G must not generate energy or hysteresis and must maintain the path-independent behavior of elastic materials. Thus, the bulk and shear moduli should be limited to the following forms (Chen and Baladi 198S)
* = *(■/,,4>
G«G(72D,e{) 3.14
Inclusion of the plastic strain tensor into the functional forms of K and G is permitted since plastic strains are constant during periods of elastic deforma- tion. Under these restrictions, the hydrostatic and deviatoric components of the elastic strain increment tensor (Equation 3.12) may be written as
df'kk dJ<
3*</,,«J) 3.15
and
***
ds<
20(i2i,.«{) 3.16
Combining Equations 3.9 and 3.12. the total strain increment tensor can be written as
A, dJx
2G d\ JL 3.17
The plastic strain increment tensor (Equation 3.9) may also be expressed in terms of the hydrostatic and deviatoric components of strain. Applying the chain rule of differentiation to the right-hand side of Equation 3.9 results in
3.18 <
> d\ a/ dJi ay, a%
a/ d^2o
which simplifies to
dtp - </x a/, "
» »f . 1 ,— —_. ,, 2 viJ2£> ar2D
3 19
22
Multiplying both sides of Equation 3.19 by bt) gives an expression for the plastic volumetric strain
Chapter 3 Tto C«j» Modei and its tmptonwntrtton
•X.
,p - 1J\ df depkk = 3d\^L 3.20 ** dJx
By definition, the deviatoric component of the plastic strain increment tensor is written as
rfeJ-Aj -%*{,*«,. 321
Substitution of Equations 3.19 and 3.20 into Equation 3.21 gives
*5--^r< 3.22
The proportionality factor dX must be determined prior to evaluating any of the above plastic strains. Baladi and Akers(1981), Chen and Baladi(198S), and Rohani(1977) outline the methods required to evaluate the proportionality factor. Those methods are included here for completeness.
Using Equations 3.4, 3.5, and 3.6 or 3.7, the total derivative of the loading function / may be expressed as
df * -LdJx ♦ ■^—%=sijdsij ♦ V -^-dtpmm - 0 3.23 °*\ -> ft a fi J J OK a,P 2SJ2D °yJlD °lkk
This expression is known as the "consistent condition" for strain-hardening materials (Chen and Baladi 1985). Using Equations 3.15. 3.16. and 3.20. Equation 3.23 may be manipulated to give
,„,< df Gde'j df ... df df d« n3.24
1 PlD d\lJ2D ' d(kk
Substitution of Equation 3.10 into Equation 3.24 produces
«!.)ü «-£-<*.. -dep.)-±L=s V2D 3.25
3K(dtkk - dt\k)ZL * —Z—{deu - de')—Z 1 Jim OM.
'J
- I^X bf df d" dt"kk
Cnaptv 3 ttm Cap Mod«! «nd Kt tmptemwttMion 23
t ByaaaB|iBi|i|fiH|fija{|
By substituting Equations 3.20 and 3.22 into Equation 3.25, one obtains
3KJHdtkk + -° *L-sudeu
Vy2£> d v-7: 2D 3.26
d\ 9K dJx
2
♦ G f 3/ 1 dp2D
^2 - 3Ü£iZ_£i
3ii d* A.P 1 OK 3e **
Solving for the proportionality factor d\ yields
d\ *
3KlLdtkk ♦ _£_ H—s de —— * ijuc ij
2D
9K dJx
3.27
♦ G *f ^2
b{l 2D
? df df dK
dJl d* dtp kk
By using Equations 3.17, 3.19, and 3.27, the total strain increment tensor may be written as
dJ, dS:: dt.. « 15 ♦ '1
u 9K >J 2G
3K^Ld(kk ♦ 9f
p2D d P\ Vv
ID
2 piD * \r;
? df df 8K
dJ, du Atp 1 °ekk
3.28
24 Chaptar 3 Th# Cap Modal and It» tmptamantation
The stress increment tensor may be written as
doiJ'KdUktiJ*2Gdeu
9K
3*Ä<* kk Bf
piD b vy: ID
sUdeU
♦ G 9f
bfl ID
3 df Bf OK
1 dtkk
3.29
irVi. + G *f ...' 1 v^o ^V^D
Equations 3.28 and 3.29 are the general constitutive equations for an elas- tic work-hardening plastic isotropic material (Chen and Baladi 1985). To use these equations, one must first define the loading function /, the functional forms of the elastic moduli K and G, and the hardening parameter K for the material of interest.
Elastic-Plastic Constitutive Matrix
In the following section, the equations for the elastic-plastic constitutive matrix are formulated. The equations are written in matrix format to render a more compact form of the equations. The development of the elastic-plastic constitutive matrix follows the derivation of Owen and Hmton (1980).
The loading or yield function / for a general work- or strain-hardening elastic-plastic model (Equation 3.2) may be rewritten as:
/(*.«) - F(0) - *(«) -0 3.30
where (in matrix format) » is the vector of normal and shear stresses and * is the hardening parameter that controls the expansion of the yield surface. Recall that in the cap model, the cap itself is the only strain-hardening yield surface; the failure envelope is not a hardening surface. Equation 3.30 may be differentiated to give:
df >ld..dJ. d« - 0 3.31
or in another form:
Chapw 3 Tht Cap Modal and Its Implementation 25
a1 do - Ad\ = 0 3.32
26
where
aT = *L - -1L 3.33 3a Boy
and
A--'*Ld* 3.34 flX OK
Owen and Hinton refer to the vector a as the/tow vector. The sealer 4 will be identified as the plastic hardening modulus.
The total strain increment tensor (Equation 3.10) may be written in matrix format as
dt *dt' + dS 3 35
By substituting for both the elastic and plastic strain increments, i.e., using the matrix equivalents of Equations 3.9 and 3.11, the following expression is obtained:
dt - D'ldo ♦ d\*L 3.36 09
where D is the matrix of elastic material constants and the inverse of the material response function
After multiplying Equation 3.36 by aTD one obtains:
aTDdt - a7 do ♦ aTDad\ 337
which may be refined further by eliminating aT do with the use of Equa- tion 3.32 to produce:
aTDdt *\A * aTDa }d\ 3 38
This leads to an expression for the sealer term dk:
d\ - *TDdt 3.39 \A * aTDa]
This term gives the magnitude of the plastic strain increment vector and is the matrix form of Equation 3.27. Note that in the text by Owen and Hinton (1980), this expression was printed incorrectly.
Chapt«f 3 The Cap Modal and Its trnptemamation
•
Having defined an expression for dk, it may be substituted into Equa- tion 3.36 to give:
D'ldo = 1 - dT
Da
A + dDa
dt 3.40
where dD - a' D.
Multiplying both sides of Equation 3.40 by D gives:
do D Dad',
A + dDa
dt 3.41
which is an expression for the elastic-plastic incremental stress-strain relation.
If we substitute dD = Da, then the elastic-plastic constitutive matrix may be
expressed as:
Dtp *D*D
A ♦ dna
3.42
Plastic Hardening Modulus
The plastic hardening modulus A must now be evaluated for a strain-hard- ening formulation such as the cap model. If the hardening parameter K is a function of the plastic strains, i.e..
« ' gUp) 3.43
then Equation 3.43 may be differentiated to give:
du » —dtp 344
Chapttir 3 Th» Cap Mod») and Ks tmptwntnution 27
28
Substituting Equation 3.44 into Equation 3.34, substituting for tp, and rear- ranging produces:
A - - V Jl *£ 3.45 OK d(PBo
The plastic hardening modulus A will be dependent upon the functional form of the loading or yield function / and the hardening function used in the cap model.
Cap Model Implemented into JAM
The following section describes the version of the cap model implemented into the finite element code JAM. The functional forms of the equations are outlined, and the cap model is described in more detail.
Two elastic response functions govern the behavior of the model in the elastic regime. The elastic bulk modulus is defined by the following equation
K(Jltt"kk) P \ , K' 1 -AT,
1 - K.cxpiK^) 3.46
where K( = the initial elastic bulk modulus and
tf i and K2 are material constants. The elastic bulk modulus prescribes the unloading moduli in pressure-volume space. The three material constants may be determined from the unloading data obtained during hydrostatic loading tests. The elastic shear modulus is defined by the following equation
G(J2D.tpkk) G
1 P 1 -C,ap(C2«;4) 3.47
where Gt - the initial elastic shear modulus and
C, and G2 are material constants. The elastic shear modulus prescribes the unloading moduli in principal stress difference-principal strain difference space. The three material constants may be determined from the unloading shear data of triaxial compression tests. The constants for the elastic bulk and shear moduli may also be determined from uniaxial strain compression tests, i.e., from the unloading slopes of the stress path (* 2GIK) and stress-strain curves ( - K * 4/3 G).
In the current model, the failure envelope portion of the loading function / is defined by a modified Drucker-Prager failure surface (see Figure 3.1) of the form
Chap!« 3 Tht Cap Mod«! and It* Imptamamatton
1 1 2 £ L
j lÄ
'"^^j fSliptk Cap
Cap J TranMtian jf
XM - K«>
R
1
•m— LM —■> - »ri - Ui) — -Ji
•a — Ki> ■»
3.1. Cap model yield surfaces
A(i, ,/^T) • /^" - [i4 - Ccxp(Byj)] if 7, >L(<) 3.48
where X, B and C are material constants. These constants may be determined from the locus of triaxial compression failure data plotted in the appropriate stress space. The strain-hardening yield surface or cap is described by the following
HUi.ftn.*)-^ 2D 3.49
- ~{[X(x) - L(K)}2 - [7, - LU)]2}05 if J, <L(K)
where X{ *) and H «) define the values of 7, at the intersection of the cap
with the Jj axis and at the center of the cap. respectively (see Figure 3.1); * is the hardening parameter, which is equal to the plastic volumetric strain, i.e..
Chapter 3 Tha Cap Modal and in Imptamarwatton 29
^ 3.50
Equations 3.48 and 3.49 also indicate the value of Jx determines which of the
two yield surfaces should be used. R is the ratio of the major to minor axes of the elliptical cap and has the following functional form
R[L(K)) =/?, +R, 1 -cxp(R2{L(K) ~L0}) 3.51
where Rt■., Rlt R2 and L0 are material constants; L0 defines the initial location of the cap.
Chen and Baladi (1985) explain that the functional form of the cap was chosen such that the tangent of its intersection with the failure envelope is horizontal. This condition is guaranteed by the following relationship between XU) and LU)
X(K) * L(K) - Rh(l(K), fin) 3.52
where
U*) { /(«) if /(«) < 0
0 if /(*) «s 0 3.53
The hardening function for tkh model is defined by
3.54
which may be rewritten in the following form
X(K) In Ji ♦ 1 W
3.55
where £>. W and X0 are material constants. W establishes the maximum
plastic volumetric strain the material can develop; X0. like LQ, defines the initial location of the cap.
As described previously, Drucker's stability postulate places limits on the functional forms of the equations in the cap model Sandier and Rubin (1979) specify some of those limitations, (a) Q( J,) must decrease monotoiücally
with increasing values of /3; (b) to avoid work-softening, the functions
30 Chapter 3 The C*p Model and Its lmptem«nt»!ion
•2.
X( K) and L( K ) must be continuous and monotonically increasing functions and
ifäO and *Z<0 ay, a«
(c) the cap must extend from the J{ axis to a point on or below the failure envelope h, i.e.,
F[X(K),K) - 0 356
and
F[L(K),K] *Q[L(K)) 3.57
(d) within the yield surfaces defined by
fi^~ <Q(JO for/, >L(K) 3.58
and
//HT <F(■/,.«) forL(«)^y,sX(/c) 3.59
the material response must be isotropic elastic. Sandier and Rubin (1979) explain that if the inequality in Equation 3.57 is true, then a gap exists be- tween the cap H and the failure envelope A and a von Mises type failure sur- face is used as a transition between the two yield surfaces (Figure 3.1). The yield surface for /, al( K ) is thus defined by the following expression
fin -min{F(KO.«l . QW) 3.60
In the evaluation of plastic hardening modulus A. one need only be con- cerned with the functional form of the cap since it is the only hardening sur- face in the model. In order to evaluate A, each of the three terms in Equation 3.45 must be evaluated. Recalling Equation 3.50, the first term may be ex- panded as
dW d//aX dH dX dXdi TXz.r dt tk
3.61
;najw«# 3 Th« Cap Mod«) and lit impiamamaiion 31
mm mmmmlm» mjmmmmw^mmmmt
32
A^Laj
Each of these terms is evaluated as
dX
ami
Kk D[(pkk * W
dH
3.62
2(L-X) 3-63
dX
Combining the terms gives
dH . 2(1 -X) d" DlSkk + W
The second term in the expression for A is evaluated as
-^ ■»«
since « ■ e£t. The fmal term in A can be expanded as
HL-OLt + til di d0" W> " 2{J^dfl
3.64
3.65
dH dH6j + *»_ 9H_ 3 M
2D
Recalling that JI;6(; - 0 and substituting Equations 3.64-3.66 into Equa- tion 3.45, one gets
Am 6(X- L) dH A - -_ 3.67 D{tpkk ♦ W)dJ\
as an expression for A. Simplifying this further by evaluating
|£ - 2Ul - L) 3.68 dJi
and substituting into Equation 3.67, one obtains the final expression for the
Ch*(K«r 3 Tht Cap Mode» and Ht Implemenutton
plastic hardening modulus
121 A - —
DU»kk ♦ W)
12(X - L)(7, -L) /I . _J Hi 1 3.69
Numerical Implementation of the Cap Model
Introduction
The numerical algorithm for the cap model was published by Sandier and Rubin in an attempt "to facilitate the general use of the cap model in dynamic computations, as well as in model fitting'' (Sandier and Rubin 1979). The cap model algorithm was designed for use in either finite element or finite differ- ence codes and is applicable to both static and dynamic problems (Chen and Baladi 198S). Of notable foresight on the part of the designers was their use of function statements within the model, which allow substantial changes to be made to the cap model's potential functions with little programming effort. This feature has allowed investigators to simulate a wide variety of natural and man-made materials with high degrees of fidelity between model and material response. Despite the many published variations of the cap model, the original cap model algorithm developed by Sandier and Rubin still forms the foundation of most current cap model algorithms.
The cap model algorithm is essentially an implementation of Equa- tion 3.29. To march the calculation through time, the user must input the
stresses 'atJ and the location of the cap at time t, which is explicitly defined
by the term /('* ) and implicitly defined by the hardening parameter '«, and the strain increments from the solution of the field equations for the current
time step '*A'<ft,;. The cap model returns the new stresses "^'o^andthe
updated cap location and hardening parameter /('**'* ) and '**'« at time t* At. A given strain increment may invoke four different types of stress paths that coincide with four different algorithms within the cap model itself:
(a) an elastic algorithm. (b) a failure envelope algorithm, (c) a hardening cap algorithm, or (d) a tension cutoff algorithm.
In the following text, a description of the four numerical algorithms is provid- ed The descriptions are basri upon previous descriptions by Baladi and Akers (1981), Chen and B»'*di (1985), Sandier and Rubin (1979), and Meier (1989). To simplify the p station, a description of the cap models re- sponse in the tensile regime will be deferred to the later pan of this section.
Chapter 3 It» Cm Mod«! «rat ttt tmpi«m«ntation 33
-™
34
Elastic algorithm
To start the numerical procedure, it is assumed that the given strain incre- ments produce an entirely elastic stress path. A set of elastic trial stresses are calculated from
and
%= 'Sij + 20'^deij 3.71
The elastic trial stresses are tested with respect to the tension cutoff, the fail- ure envelope, and then the cap. If these surfaces are not violated by the trial stresses, the actual stress path is an elastic path, and the new stresses are the
'♦Aff _£, «^J MA». _£. elastic trial stresses, i.e., ' aiJy -CJ{ and 5» * s «;•
Failure anvelope algorithm
If the following conditions exist when the elastic trial stresses are tested with respect to the failure envelope,
h{EJx,(%^) -/T/~- Q(EJX) ± 0
then the elastic trial stresses have violated the failure envelope, and the given strain increment must be a combination of elastic and plastic strains. The trial stresses must be corrected such that (a) the final stress state falls on the failure surface and satisfies the following relation
»('•AVV "*>JW ) . 0 3 n
and (b) the resulting elastic and plastic strain increments add up to the given
strain increments ' * A'rf« ^.
The mathematical statement that requires the final stresses to lie on the fixed failure surface is given as
dh - 4^-d"h * ° 373 BotJ >
Assuming small strain increments. Equation 3.73 can be numerically approxi- mated by the following expression
Chapter 3 The Cap Modal and Its Imptamamation
A
dh (%D~ ~ {'^'hD ~ Q( %) * ß( "A,'i> 3.74
which reduces to
dh *{%D -Q( */,) 3.75
since the final stress point must lie on the failure surface, i.e.,
/ t + Ati
'2D t + At ß('♦<"/,)-<>
3.76
Equation 3.75 may be substituted into Equation 3.73 and expanded in the following manner
(%D -Q(EJO'-^-äaij
>J
dh *J\ dJ\ 9aij
dh dp2D
dp. 2D
11 dJ,
dJx
da.
dh E
dOjj 3.77
2SJ2D d P:
sijdsiJ 2D
where dJl * EJ\ - '7, and ds^ * Esij -'sy. From Equations 3.70 and
3.71, we know that dJx « 3K"*'dekk and dsi} - 2G'*^deijt and these expressions may be substituted into Equation 3.77 to give
f '2D -Q(EJx)-lK,tl"d(kk*!L OJ \
plD d VJ2D
dh £s„ ""de*
3.78
ChafKt» 3 Th» Cap Modt* and tu lmp»amam»t«n 35
a^mmiSm •£■ _•_
If we substitute Equation 3.78 into the numerator of Equation 3.27 and recog-
• .1, ♦ u f 9h dQ mzethat A=/, —- - --f,
3y72D ure envelope, an expression for d\ may be written as
Bh = 1, and — - 0 for the fixed fail- die
dX m /%T -Q(EJQ
9K dQ 3.79
♦ G
Substituting the above expression into Equation 3.20, the final expression for the plastic strain increment is obtained
*i* -3
■ "
i%D -Q(EJO
9K
2
♦ G
dQ 3.80
t + At An expression for a'J{ may be developed in the following manner
t*Ali V 'y, ♦ 3K'**'dtkk - 3Ar</«Jt
£J, - 3Kdtp
kk
3.81
where d«J4 is defined by Equation 3.80. A "tentative" value of '*A'y, may be calculated from Equation 3.81; this value is tentative because it must be tested against the current position of the cap, which is defined by the value of
L('K).
If '*A'Jl < L('K), which indicates the stress point has violated the cap, then comer coding is required, i.e.. the cap must intersect the failure envelope
forming a comer, and the value of '*4,i, must be adjusted. Adhering to the imposed conditions of normality, a stress state lying on the failure envelope produces dilatant plastic volumetric strains. Since cap expansion can only result from compressive plastic volumetric strains, the cap is stationary, and the new stress state can not move beyond the intersection of the cap and the
failure envelope. Thus, the final stress state is "A'7, * £('*). and the up-
dated hardening parameter is '*A'* ■ '*.
36 Chapur 3 The C»p Model and Its IfnotemantMion
JL
If '+A'/j > L ( 'K ), then the final stresses will depend upon the form of the hardening function. Equation 3.7 is the simplest form of the hardening function to use because it only permits plastic volumetric compaction, i.e., the cap is only allowed to expand. As in the above case, a stress state lying on the failure envelope produces dilatant plastic volumetric strains. Since the hardening function defined by Equation 3.7 prescribes no cap movement due to dilatant volumetric strains, the cap is stationary. Thus, the final stress state
is t*A'Jl, i.e., no adjustment is required, and the updated hardening parame-
ter is '+A/„ - ' K . If the hardening function takes the form of Equation 3.6, which allows the cap to expand and contract, the cap is adjusted (in this case contracted) to a position prescribed by
t*&t, I - '/ ♦
dl
dt kk
dt kk 3.82
'/
and a tentative value of ' * A'K is obtained. The new position of the cap must t*At be compared to the value of *a\7,. If the cap has contracted such that
r*At Jx < L('*A,K) * ,+A'/,both "a'«and 'tA'J{ must be adjusted
suchthat '*A'Jl = L('*A'K)
with the following relation
l ♦£/ /. This is accomplished by starting
% -3Kdtpkk t*Ali L( I* At
K) - '/ ♦ 3/ dt p 3.83
kk
eliminating dtpkk, which is the third unknown, by substituting the following
Kk 'i
JL dt"kk
3K 3.84
'/
from which one can show that
Chapt* 3 Th« Cap Model and Its lmpt«m«raation 37
,+A'y, = EJX -3K *J ~ '<
dl + 3K
'1
EJ dl ♦ 3* - 3K(EJX - '/)
dl
d*pkk 'i
♦ 3K
3.85
which in turn simplifies to
,+A'/ »/('*A,0 - '+AV,
a/ £y, * 3K 'i
di
de"kk
+ 3K
3.86
t*At, Having calculated the final value of '*a'Jl, we must calculate the new com-
t*Atr TV. -™««J«« for '♦*', portents of the deviatoric stress tensor stJ. The expressions are developed below.
'j
Recognizing the path independence of linear elastic constitutive equations, we can write
'•%- V2Grf<J 3.87
Substituting Equation 3.71 into Equation 3.87 and performing a simple manip- ulation one obtains
l* Al, !s-" *■$ - 2Gde' 3.88
Recalling Equation 3.22 and recognizing that „„,,„£. ,.. - 1, we can write
de' - d\
ij p== * ij 3.89
38 ChaoMr 3 Th» Cap Model and Its tmptemcnuuon
timmmiamm^gllmmm
which can be substituted into Equation 3.88 giving
a ij J ij d\G i* At.
/ 1+Alj
'2D
After rearranging the above equation one obtains
/♦A», IJ
1 ♦ d\G
I r + A/r j- ID
3.90
3.91
Squaring each side of Equation 3.91 and multiplying by xh produces
2 t*Ali
'2D 1 ♦ d\G
f**'J. 2D
'2D 3.92
Taking the square root of each side and rearranging terms, one obtains
d\G / »♦An
'2D
f 1 ♦ 3.93
'2D / /♦An
'2D
Replacing the right-hand side of Equation 3.93 with the expressions in Equa- tion 3.91, one obtains
/ »♦A*/
J2D /♦An
f '2D 'ij
which may be rewritten for our use as
»♦Al. / »♦Ali
'20 Er
fEj- 2D
to calculate the new dcviator stress tensor components.
3.94
3.95
Chapter 3 Tht Cap Modal and It* tn^tamanutton 39
J. mmammämmmmm ■iM '■ — EiA
Cap algorithm
If the failure envelope is not violated by the elastic trial stresses, the trial stresses are checked against the loading function for the cap. If the following conditions exist
and
or
H(EJl,^EJ2D ,'«) > 0
EJX < X('K)
EJi £ L('K)
then the cap algorithm is invoked and the position of the cap is adjusted until
ff('+AV/ /♦A* r 2D
An iterative procedure is used in the cap algorithm. To start the proce-
dure, a trial value of dl(l) is assumed in order to calculate a new trial cap
position '*A'/(,) = /(,) « '/ + dl(i\ where the superscript i denotes an
iterative value. In addition, trial values of K(,) ,£(K(I)) ,X(»C(,)), and
dtpkk are computed. Finally, a trial value of Jx is computed from the following relation
r(0 f/, - 3Kdep kk
3.96
If y,(0 £ X( KU) ), a smaller value of dl(i) is assumed. If
J, £ L( *(l)), a larger value of dl(l) is assumed. This process is carried
,(«)• (i) ,<•>< on until the condition I(«1") S7,v" <X(*U') is satisfied. The final value of / is one which satisfies the following equation to some desired accuracy
{ «♦A/j 'ID
Gdi
3{ ** -/< '20
3.97
40
where
y1(".«(',
dF
j/0./"»
The derivation of Equation 3.97 is outlined in the following text.
3.98
Chapter 3 The Cap Model and its Implementation
A
If we start with Equation 3.22 and substitute for d\ using Equation 3.20, one obtains
deij °sijd*kk 1 8H
6v72D drw
,dH 37,
3.99
Recognizing that / » H, BHIdp1D = 1 and bHldJx = -3F/dJl = £, we can rewrite Equation 3.99 as
defj « s dep
kk ,J r== 3.100
Substituting the above into Equation 3.88 and rearranging gives
1 ♦ Gdep
kk
3*\ST -Es 'J
3.101
Performing the same operations on the above equation as was used on Equa- tions 3.91-3.93, one obtains Equation 3.97.
The solution of Equation 3.97 is obtained through the use of an iterative convergence routine known as the modified regula falsi method (Sandier and Rubin 1979). A dimensionless function P( I) is defined as
/»(/)
/(«) -/ (i)
/(«) -X(K)
/^T-/ f*Al/(«) J2D Gd*'u
TT
/%T*/ l*Al|(0 3(
if y,(,)sX(«)
if X(K)<J,U) <L(K)
U) X(K) -J
L(K) - X(*) if y,(,)^L(K)
3.102
where the solution P (I) - 0 is also the solution of Equation 3.97. If we can
show that f/, < '*A'l(<) < '/, then P(/) is bounded and monotonic in the
strict sense, and the solution /*(/)■ 0 is unique and can be found to any
Chapter 3 Tha Cap Mode! and Its Imptomamatton 41
42
desired degree of accuracy (Sandier and Rubin 1979). An expression for the degree of accuracy or tolerated error is given by
VhD "/ f + A/i "att* "7 Gde P
2D 3* 3.103
< NQ[X(K)}
where a tolerance of Nl h ( » , Jj2D ) = 10 3 (in dimensionless format) is typically used.
To show that EJX < '*A,l(l) < '/, we must recognize that /,(l) < L ( 'K ) must be true, since it is a condition for invoking the cap algorithm. In addi- tion, since
/<'> = £y, -3Kd*pkk
3104
we know that /, > EJl, because the plastic volumetric strain increment is negative during volumetric compaction, i.e., when the cap expands. This
means that the final value of ,*AtJl must lie in the range EJX < '*AtJx <L{'K)
Now let us determine the lower limit of '*AtJx , which will lead us to the
lower limit of '*A'/('). The cap exhibits its furthest expansion when EJX is at the intersection of the cap and the failure envelope. When the cap is in this
position, '*A'JX * L('*A'K) ■ EJX, which implies that the lower bound of 'l
'♦*'/<'> is
'*A'/ - /('♦*'«) -t(f*A,K) - EJ
The upper bound of '*A'l(,) is simply the value at time t, i.e., '*A'l * '/.
Combining these expressions, the range of '*A'l(,) must be EJX <<**'/(,) <'/
With the above conditions satisfied, the solution to Equation 3.97 may be obtained. This concludes the description of the cap algorithm. A description of the cap model's response in the tensile regime follows.
Tensile algorithm
Sandier and Rubin (1979) recognized that soil tensile data is seldom ob- tained in the laboratory and therefore dealt with tensile behavior in a simple manner. They also cautioned potential users of the simplistic nature of the cap model in the tension regime. A tension failure response is invoked if
Chapter 3 Tha Cap Modal and Its Implamantation
EJl > T, where T is the tension cutoff or limit. Sandier and Rubin recom-
mended that the final stresses be defined as ,+A'/1 = T and '+Atstj = 0 when the tension cutoff is exceeded. For materials using the definition of the hardening parameter defined by Equatior 3.6, the plastic volumetric strain is defined by
< - „u . ^L_3 3..05
and an updated hardening parameter is determined. If the tension cutoff is not
exceeded but EJX > 0, i.e., the elastic trial stress still lies in the tension regime, the stress state must be checked against both the failure envelope and the von Mises transition using the following inequality
iEhü * min(ß(£y,),F[L( '«),'« j}
Stress states violating the von Mises transition must be returned to that surface using the same logic implemented for the failure envelope. Stress states lying on the von Mises transition will produce no plastic volumetric strains due to the imposed normality conditions. In addition, the von Mises transition is fixed because the cap hardening surface does not expand.
Implementation of Cap Model into JAM
Two basic operations that are associated with elastic-plastic material mod- els must be performed in most implicit finite element codes: (1) the construc- tion of the elastic-plastic constitutive matrix and (2) the calculation of the residual forces. The purpose of this section is to explain how these two oper- ations were affected by the implementation of the cap model. The later opera- tion will be considered first since it is a straight forward process.
After the strain increments at tins? t * At are obtained from the solution of the field equations, the stresses at time / ♦ At in each element are calculated. The residua] forces at time t * At are then calculated based on the stress states in the elements. In this operation, no substantial changes are required in the cap model subroutines; hence, the implementation is rather simple
However, three components, the elastic constitutive matrix D, the plastic hardening modulus A, and the flow vector a, are required to calculate the
elastic-plastic constitutive matrix D'p. The calculation of the elastic constitu- tive matrix is simple and needs no further discussion. In JAM, a modified cap-model subroutine calculates and returns values for the flow vector and the plastic hardening modulus. This subroutine first determines which of four possible regions or surfaces a stress point resides in or on, i.e., an elastic
Chapttr 3 Th« Cap Model and Its Imptamantation 43
mmmmmmm
44
region, the failure surface, the cap, or the tension cutoff. The plastic harden- ing modulus is nonzero only when the stress point falls on the cap since it is the only hardening surface. In this case, the plastic hardening modulus is calculated using Equation 3.69. The flow vector is calculated by numerically evaluating Equation 3.66 when the stress state lies on the failure surface, the cap, or the tension cutoff.
To complete the implementation, one must provide the model access to the material constants and an array to store the location of the cap for each nu- merical integration point.
Verification
To insure that the cap model was correctly incorporated into the finite element program JAM, several laboratory stress- and strain-path tests were numerically simulated. These calculations were compared to the output from a cap model driver (Chen and Baladi 198S) exercised over the same laboratory stress and strain paths. The two programs should produce similar if not identical results.
The following tests and strain paths were simulated: a hydrostatic compres- sion test with one load/unload cycle, a set of constant radial stress triaxial compression tests, a set of constant mean normal stress tests, a uniaxial strain (K0) test with one load/unload cycle, and finally a test with a K0 load/constant axial strain (BX) unload cycle. To simulate the tests with the finite element program, a single element was loaded under the appropriate boundary condi- tions. Several loading increments were utilized during each calculation, and a convergence tolerance of 1 percent was satisfied at the end of each increment. The output at the end of each increment is represented by a symbol on the comparison plots.
The simulated hydrostatic loading test consisted of an applied loading to a pressure level of 250 MPa, followed by an unloading to zero pressure. This calculation exercised the logic and code affecting both the cap movements and the elastic algorithms within the model and finite element program. The finite element and cap model driver results are compared in Figure 3.2. Output from the finite element program matches the cap model driver with no notice- able errors.
Four constant radial stress triaxial compression tests at radial stresses of 25, 50, 100. and 150 MPa were simulated. Loading was terminated prior to reaching the ultimate failure surface, anJ unloading results were acquired for only three of the four calculations. The values of principal stress difference calculated by the finite element program were less than those of the cap model driver (Figure 3.3). The magnitude of the errors decreased when a larger number of increments was applied.
Chapter 3 The Cap Modal and It* Implementation
:igure 3.2. Simulated hydrostatic loading stress-strain response
100 -
i '° /
a
o
it
a *:<
ii DWi 1 © 1
F j 1 1 1 1 1 ^ 1 1,,,, Jl • .' > 4 " fc
vtB" ( ;*, i'fc* '. PJB'f'.*
Figure 3.C 1. Simulated triaxial compression stress-strain response
Four constant mean normal stress tests at confuting pressures of 25. SO, 100. and ISO MPa were simulated. As with the simulated triaxial compres-
Chaptar 3 Tha Cap Modal and tts tmpiamantation 45
•2.
100 -
a 3
90 f OJ
y
a LJJ
a
60
in
a 40
a
L1
EE a 2 0
i ■ i x _ i 0 t ; 3
VCPTlCA; STQAIN PERCENT
6
Figure >3.< 1. Simulat ad stress-strain respon se of constan t moan nt »rmal stress tests
sion tests, loading was terminated prior to reaching the ultimate failure sur- face; unloading results were not acquired. The finite element and cap model driver results are compared in Figure 3.4. Errors in the stresses calculated by the finite element program were less than those in the simulated triaxial com- pression tests (Figure 3.3).
The simulated K^ test consisted of an applied loading to a vertical strain level of 20 percent, followed by an unloading to a small value of vertical stress. In this calculation, the finite element simulation was conducted with a displacement controlled boundary condition. This type of loading should produce an exact match between the finite element program and the cap model driver since no strain increment iterations are required in the finite element program. The calculated K<, stress-strain response is plotted in Figure 3.S and the stress path response in Figure 3.6. There are no noticeable differences be- tween the two calculations.
A KQ load/BX unload test was also simulated with a displacement control- led boundary condition. The test consisted of an applied loading to 20 percent axial strain, followed by an unloading to a small value of radial stress (Fig- ure 3.7). In this calculation, the corner coding of the cap model was exer- cised as the stress path unloaded along the failure envelope (Figure 3.8). The calculated results suggest a proper implementation of this logic.
46 Chapm 3 Th§ Cap Mod*« «Ml ft» imeteAMmauon
100 r
BO
10
in UJ a
< / II ^ 40
20
/ lilt i i i i t *; 10 15 ?0 ?5 30 r.
VERTICAL STHAIN PERCENT
Figure 3.! S. S*'ess-strain response of simulated UX test
Figure 3.6. Stress path from simulated UX test
Chapter 3 Tha Cap Modal and I» hnptamantanen 47
iMm ^^~
Figure 3.7. Stress-strain response of simulated UX/BX test
Figure 3.8. Stress path from simulated UX/BX test
48 Chapter 3 The Cap Modal »nd its Imptcmcnutton
Summary
The features of the cap model and the relevant equations were documented in this chapter. In addition, the steps required to implement the cap model into the FE code were summarized. The implementation of the cap model was verified by comparing the output from die FE code and a driver for the cap model.
Chapter 3 The Cap Modal and its tmpt«m«nutK>n 49
4 Equations of State for Air, Water, and Solids
Introduction
The materials of interest to this investigation include partially-saturated soil- and rock-like geomaterials and man-made concretes, grouts, and grout- cretes. As outlined in Chapter 2, the equations developed from the Biot theory require an expression for the bulk modulus of die pore fluid ami the grain solids. To determine the bulk modulus of the pore fluid, the concept of a homogeneous pore fluid will be adopted to treat partially-saturated materials. This investigation will assume that the liquid within the pore space is water and the gas within the pore space is air. Thus, the pore fluid will be regarded as a compressible mixture of air and water. Based on the equations of state (EOS) for air and water, we will develop equations for the bulk modulus of this air-water mixture. The grain solids «ill be treated as either linear or nonlinear elastic materials or as nonlinear hysteretic materials; each method for calculating the bulk modulus of the grain solids will be described.
Equation of State for Water
Over the pressure range of interest to this investigation, i.e., 0 to 600 MPa, water has a finite compressibility and should be treated as a nonlinear elastic compressible material. The compressibility of water is depicted in Figure 4.1 as a plot of pressure versus volume strain. The reader should note that at 600 MPa the volume strain of water is nominally 14 percent. The bulk modulus or compressibility of water was evaluated from the Walker-Sternberg EOS for water (Walker and Sternberg 1965). which is valid for pressure levels of up to SO GI*a. The EOS expresses the water pressure as an analyti- cal function of the density and the internal energy of the water. For this investigation, the energy dependent terms in the EOS were not included due to the assumed quasi-static and isothermal nature of the intended calculations. Without the energy terms, the EOS is expressed as
50 Chapter 4 Equations of State for Air, Water, and Solids
600 /
500 /
400
i 3 100
• a.
?00
100
0 2 * 6 8 10 1? 14
Votum Striin, ft
Figure 4,1. Pressu re versus volume strain response of water
P - P/I + P3/2 + P5/3 + />7/4 - J»c 4.1
where P is the pressure in the water, p is the density of the water, /, are
material constants and PQ is the initial pressure. If we define volumetric strain as
-** i-!2 4.2
then the bulk modulus of water may be expressed as
„ dP p2 dP PO rfP rfe
4.3 kk
Substituting Equation 4.1 into Equation 4.3, one obtains the final expression for the bulk modulus of water as a function of density
'.'-(p2fi *3p4/2 -5p6/3-7p8/4 Po
4.4
In the FE program, pressure is the known quantity, not density. Since the EOS expresses pressure and bulk modulus as a function of density, Newton's method was used to calculate the density for a given pressure, then the bulk modulus was calculated.
Chapter 4 Equations of Slate tor A», Waw. and Solid* 51
52
Air-Water Compressibility
Background
The concept of a homogeneous pore fluid (HPF) was first introduced by Chang and Duncan (1977). In using their concept, one assumes that a three-phase material containing air, water, and solids may be replaced with a two-phase material containing a compressible pore fluid and solids. A partial- ly-saturated material is transformed into a fully-saturated material with a HPF. Effective stress is calculated in the same manner as for a fully-saturated mate- rial, and the modulus of the pore fluid is calculated based on the modulus or compressibility of an air-water mixture. The concept is applicable to materi- als with saturation levels greater than 85 percent. At these levels of satura- tion, the air should be in the form of occluded bubbles uniformly distributed throughout the water, and the air and water pressures should be identical. At lower saturation levels, one can not guarantee that the air and water pressures will be the same.
The compressibility of air-water mixtures has been studied by several investigators. Bishop and Eldin (1950) examined non-zero total-stress friction angles measured during undrained shear tests. They attributed the observed behavior to incomplete saturation of test specimens. Using Boyle's and Henry's Laws, they developed expressions for the compressibility of an air- water mixture without accounting for surface tension effects.
Schuurman (1966) reviewed the work of previous investigators and con- cluded that surface tension effects must be included in an air-water compress- ibility formulation. Schuurman claimed to be the first to attempt such a for- mulation. Schuurman assumed that at saturation levels greater than 85 percent, the air existed in the form of bubbles. However, to account for surface tension, the radius of the air bubbles was required, yet little if any experimental data was available to provide this necessary information. Schuurman's formulation also differed from that of Bishop in that he wrote his expressions in terms of the current volume of air as opposed to the original volume, and he assumed the water was incompressible.
Fredlund (1976) also developed an expression for the compressibility of an air-water mixture using a formulation in which the water had a finite com- pressibility. He accounted for surface tension in a manner that did not require a knowledge of air bubble sizes by using a parameter for air-water pressures similar to Skempton's B parameter, which could be evaluated experimentally. Fredlund also interpreted the mixture volume in the expression for compress- ibility as the volume of water plus free air as compared to water plus total air.
Chang and Duncan (1977) based their expressions for the compressibility of an air-water mixture on the equations of Schuurman. Like Schuurman, they included surface tension effects in their formulation and assumed the water was incompressible.
Chapter 4 Equations of State (or Air. Water, and Solids
Alonso and Lloret (1982) reviewed the work of previous investigators, compared the compressibility curves of each, and formulated their own ex- pressions for the compressibility of an air-water mixture. They assumed a finite compressibility for water and accounted for surface tension in the same manner as Fredlund.
In summary, there are significant differences in the equations developed by several investigators for air-water mixtures. For this reason, equations for the compressibility of an air-water mixture will be developed in this chapter. Prior to developing the equations, a brief description of the appropriate physi- cal laws will be provided.
Boyle's and Henry's laws
Boyle's and Henry's Laws will be used in developing equations for the compressibility of an air-water mixture. These laws are defined and explained (or purposes of completeness. Boyle's Law states that "at a constant tempera- ture, the volume of a given quantity of any gas varies inversely as the pressure to which the gas is subjected" (CRC Handbook 1980).
Air dissolves in water according to Henry's Law, which states that "the weight of gas dissolved in a fixed quantity of liquid, at constant temperature, is directly proportional to the pressure of the gas above the solution" (Fredlu- nd 1976). Fredlund (1976) explains that the structure of water molecules pro- duces a "porosity" within the water of approximately 2 percent by volume. This porosity can be filled by a gas such as air, i.e., air dissolves in water by filling this pore space (see Table 4.1). Fredlund (1976) provides a simple analogy to understand the compress- ibility of an air-water mixture.
Table 4.1. Solubility of Air in Water
Consider a test vessel made of a cylinder and piston. At the base of the cylinder is a porous stone having a porosity of 2 percent; the porous stone simulates the behavior of the water. The piston is initially posi- tioned some distance above the stone with air filling the space in between. An imaginary valve at the surface of the porous stone controls the move- ment of air into the stone. The air in the porous stone simulates the air dis- solved in water. If the valve is closed and the piston moves down into the cylinder, the air above the stone com- presses following Boyle's Law If the valve is opened, some of the air will diffuse into the porous stone following Henry's Law. This process will con- tinue until all of the air passes into the porous stone When the piston con-
Temperature Degrees C
Henrv's Constant
0 0.02918
4 0.02632
10 0.02284
15 0.02055
20 0.01868
25 0.01708
I 30 0.01564
i from Fredlund (1976)
Chapter 4 Equations of State for Air, Water, and Solids 53
tacts the porous stone, there is a discontinuity in the compressibility of the system; the compressibility jumps immediately to that of water. The level of saturation within an air-water mixture must be evaluated to determine the discontinuity point.
Derivation of equations
The following assumptions were made for this analysis. We will assume initial saturation levels are greater than 85 percent, which implies that all air bubbles are occluded. Surface tension effects will be neglected, which allows us to assert that the air bubbles within the water will be at the same pressure as the water. The air is soluble in water and observes Henry's Law, and the rate of increase in pore water pressure from any simulation is slower than the rate of diffusion of air in water. Finally, prior to full saturation, the com- pressibility or bulk modulus of water is a constant. We will first develop the equations for an air-water system with a rigid porous skeleton, then one with a compressible porous skeleton.
The following terms are used in the derivation of the compressibility of an air-water mixture. Let
V denote the total volume of air and water, Vv the volume of the void space, Vw the volume of water, Vd the volume of dissolved air, Va the total volume of air, V'a the volume of free air, which is equal to Va - Vd, Pw the pore water pressure, Pa the pore air pressure and H Henry's constant.
A subscripted "o" is used to indicate an initial value. The total mass of air and water remains constant. Substituting expressions for porosity (n) and sat-
uration (5), the initial volumes of water Vw0 and free air Vao may be expressed as
* S0 Vv0 % v, s0 4.5
and
V„"<» -S0)VV0 = n0V0(l -S0) 4.6
Using Henry's Law and Equation 4.5, the initial volume of dissolved air may be expressed as
Vd0- Ko" -»oVoS0H 4.7
The sum of Equations 4.6 and 4.7 yields an expression for the initial total
54 Chapter 4 Equations of State loi Air. Water, and Solids
volume of air in the system
Vao = n0V0(l-S0+S0H) 4.8
Expressions for the compressibility of water and an air-water mixture may be written as
r - 1 dV» v-ui ~ -rr- 4.9
Vw dPw
and
"m V + V ra Tw
dPw dPw
4.10
respectively. Substituting Equation 4.9 into Equation 4.10 one obtains
cm = V + V
- V c dp-,
4.11
We will now use Boyle's Law to develop an expression for the derivative in Equation 4.11. Boyle's Law may be written as
V P = V P Ja'a 'ao'ao 4.12
If we assume the pore and air pressure are equal and Vd = Vdo, we can write the following
P P V + V 'ao ' wo Ta d
P" Pw Vao + Vd
4.13
from which we write
dIz dv'
ao d P 1 ■ \2 ' ao
k ♦l)
V 'ao P ' ao
4.14
Substituting the latter expression in Equation 4.14 into Equation 4.11, one
Chapter 4 Equations of State for Air, Water, and Solids 55
obtains
'm V + V V P Tao ' ao
+ v»cw 4.15
which is an expression for the compressibility of an air-water mixture. By judiciously substituting for the volume terms in Equation 4.15, we will devel- op a final expression for the compressibility of the mixture.
By combining Boyle's Law ( Equation 4.12) and Equation 4.8, we may write an expression for the current total volume of air
ao ya--^n0V0(l-S0 + S0H) 4.16
The current volume of water may be expressed as
Vw = Vwo(l + CW6P) 4.17
and, after substituting for Vwo, as
V„*n0VoS0(l +CW6P) 4.18
The current volume of dissolved air, which is a function of Henry's Law and the current volume of water, is written as
Vd*n0V0S0H(l +CW6?) 4.19
Subtracting Equation 4.19 from Equation 4.16, one obtains an expression for the current volume of free air
va -»„v.! !^(l-S0 + S0H) -S0H(l+Cw6P) "a
4.20
Adding Equations 4.20 and 4.18. one obtains
V ♦ V = n V ' rfl rw "o To ao (\-S0+S0H) ♦ S0{1- H)(\ + CW6P)
4.21
which will eventually be substituted back into Equation 4.15 Combining
56 Chapter 4 Equations of State tor Atr, Water, and Solids
Equations 4.8 and 4.16, one may write
'ao
ao n0V0(l-S0 + S0H) 4.22
and, by multiplying Equation 4.18 by the compressibility of water and dropping the higher order terms, one obtains
•»v ^w ~ no "o ^o ^w 4.23
Substituting Equations 4.21, 4.22, and 4.23 into Equation 4. IS yields the final expression for the compressibility of an air-water mixture
'm ao (1- S0 + S0H) *Se(l-H)(\-Cw6P)
"(l-S0 + S0H)+S0Cw
4.24
In a similar manner, an expression for the level of saturation may be devel- oped and written as
S = V * V
S0(l + CW6P) 4.25
ao (\-S0*S0H) +S0(l-H)(\+ CW6P)
When the porous skeleton is compressible, the current void volume may be expressed as
va*K- V».-««> 4.26
where V0 is the initial toul volume of voids and solids and t kk is the effec- tive volumetric strain. Substituting the above and Equation 4.18 into the first expression in Equation 4.25, one obtains
Chapter 4 Equations of State for Air. Water, and Solids 57
mmmmmmmmmmm
s = n0S0(l-Cw8P)
V + V 4.27
% ~e kk
which is an expression for the level of saturation in a deforming porous skel- eton. An equation for the compressibility of an air-water mixture within a de- forming porous skeleton may be obtained by combining Equations 4.15, 4.22, 4.23, and 4.26 to yield
"m :**
-Z(l-So + SoH) + S0Cw 4.28
In the process of a calculation, one must first evaluate Equation 4.27. If the level of saturation is less than one, Equation 4.28 is used to calculated the bulk modulus of the pore fluid. If the level of saturation is equal to one, the bulk modulus of the pore fluid is calculated from the EOS of water, i.e., Equation 4.4.
To illustrate the response of a partially-saturated material to an applied loading, an example calculation was conducted and the output graphically
IV,
»• 1 -2
/ 8-0.9 / ,' n - 0.2 / , E - 1800 / <
- K - lOOO / '
Modulus of / • All air peroait* V f i*****"^ eruahoa out
-».« i ■ » ■ »
Figure 4.2. Pressure-volume response of partially-saturated material
58
presented in Figure 4.2. The simulated material has a Young's modulus of 1800 MPa. a bulk modulus of 1000 MPa. a total porosity of 20 percent, a
Chapter 4 Equations of Stat» for Air. Water, and Solids
X
Saturation level of 90 percent, and an air porosity of 2 percent. The material was loaded under undrained uniaxial strain boundary conditions. At volume strains less than approximately 2 percent, the generated pore pressures are negligible, and the loading bulk modulus is equal to the skeletal bulk modulus of 1000 MPa. At these strains levels, the material loads as if it were fully drained. At a volume strain of 2 percent, all of the air porosity is eliminated, and the pore fluid becomes fully saturated and begins to carry a major portion of the applied stress. At these strain levels and above, the material loads as a fully-saturated material. In addition, the pressure-volume response is nonlin- ear due to the nonlinear nature of the water.
Equation of State for Solids
Three methods for calculating the bulk modulus of the grain solids were implemented into the FE program. The first method assumes the grain solids are linear elastic materials. The second method uses an analytical EOS and treats the solids as a nonlinear elastic material. The third method uses a simple model to simulate the nonlinear hysteretic material behavior of the grains.
The first method is self explanatory; the program simply uses a constant bulk modulus value for the entire calculation. In the second method, an ana- lytic relationship between pressure and compression is developed for each material. Compression is defined as
a = f 4.29
where t is the Cauchy or engineering strain. Using solid carbonate as an example material, the pressure-compression relationship is linear below 1.2 GPa and may be written as
Pg * 0.7 M 4.30
where P is the gram pressure. The bulk modulus for carbonate may then be
written as
K =11- M):4^ *07<1-M)2 4.31
A plot of pressure versus volumetric strain for carbonate is plotted in Figure 4.3 Other materials may be simulated in an analogous manner
The third model, which simulates nonlinear hysteretic material behavior, uses tabulated curves that describe the loading and unloading pressure-volume response of the grains This model is based on the work of Meier (1986),
Chapter 4 Equations o* Stale for A». Water, and Solids 59
£ *0D -
»»'„<•>* Il>|.n «
Figure 4.3. Pressure versus volume strain response of carbonate
who used a similar model for one-dimensional, plane-wave ground shock calculations.
During virgin loading, the volume strain is computed from the current value of pressure through linear interpolation of the tabulated loading curve. A similar process is employed when unloading occurs from pressure levels at or above the lockup point using the tabulated unloading curve. When unload- ing takes place from pressure levels below the lockup point, a scaling process must be applied to the tabulated unloading curve. In this scaling process, let Pm and tm represent the peak pressure and peak volume strain, respectively, from which unloading commences (see Figure 4.4). The pressure and volume strain scaling factors are calculated as
/. P,
P - P 4.32
and
/, ■ (1 " <*)fP ♦ a 4.33
where Pt is the pressure at lock up. P, is the tension cutoff pressure, and a is an empirically determined calibration constant with values ranging between zero and unity. Knowing the unloading pressure (P,), the recovered pressure
(4P), which is the difference between the pressure »: lock up and the value of pressure on the tabulated unloading curve, is calculated in the following manner
60 Chapta* 4 Equation« et Slat« tor A*. Watt», and Solid*
Figure 4.4. Nonlirtear-hysteretic model
*P-(Pm-P,)fP 4.34
The recovered strain (A e) is computed through linear interpolation of the tabulated unloading curve. The unloading volume strain («,) is calculated by subtracting the scaled value of recovered strain from the peak strain
4.35
The bulk modulus on this unloading curve is calculated as
K, -K.ifjf.) -*, a 4.36
where Ku « AP1A t. Reloading occurs along a line passing between the
last unloading pressure-volume strain point and the point Pm, tm,
Chaptar 4 Equation« of Statt to» A«, Wat*«, and Sokdt 61
i
i Summary i
The algorithms required to numerically simulate the behavior of the three # primary constituents of geomaterials, air, water and solids, were documented in this chapter. When these algorithms are combined with an appropriate A skeletal model into the FE formulation of Biot's theory, the multikilobar response of any geomaterial may be calculated.
• i
Chapter 4 Equations of Stata for Air, Water, and Solids
^ l^i__?u_j_-j_M_1j__l^i»u_M_lULn»jiaii__l__1_i_jij^ - ■■ ■ -- — ^- . . nin -— -^^ -■ ^ - —9
5 Features and Verification of FE Program
Introduction
The objectives of this chapter are to (1) describe features in the FE pro- gram that have not already been introduced and (2) present solutions from several verification problems as proof that the program works correctly. Several features of the FE program have already been introduced. In Chap- ter 2, the benefits gained from effective stress simulations of multi-kilobar material behavior and the available material models were described. The cap mode! was documented in Chapter 3 and the equations of state of air, water, and solids were described in Chapter 4.
The following features will be presented in this chapter. A restart feature was implemented into the FE program to permit the simulation of certain laboratory tests. For example, a consolidated undrained triaxial compression test wherein the consolidation phase has drained boundary conditions and the shear phase has undrained boundary conditions requires changing fluid flux boundary conditions. A brief summary of the postprocessing procedures will also be presented. These features are described in the next section.
The final sections of this chapter document solutions from several verification problems. For each problem, the FE solution is compared to a ailable closed form or analytic solutions. These verification problems e. iblish the FE program's ability to correctly solve a variety of initial and boundary value problems.
Additional Features of FE Program
Restart feature
The restart feature was implemented for the purpose of allowing the user to change the boundary conditions at a preselected time in the calculation. A KQ/BX/STX test (acronyms defined subsequently) is an example of a labora- tory test with changing boundary conditions. This test is conducted by
Chaptw 5 N«n*tt and Verification o» fE Program 63
loading a cylindrical specimen to a desired mean normal stress level under K,, or uniaxial strain boundary conditions, unloading to a desired mean normal stress level under constant axial strain (BX) boundary conditions, and then conducting a constant radial stress triaxial compression (STX) test at yet another mean normal stress level. The K^ loading and the BX unloading phases may be numerically simulated with displacement controlled boundary conditions. However, to realistically attempt to simulate the STX phase, the user should apply stress controlled boundary conditions. The restart feature implemented into JAM allows the user to perform this calculation in a simple manner.
Postprocessing
In many instances, one would like to plot FE results at a single location, e.g., at the nodal points. Many postprocessing FE software packages require stress and strain values at the nodes rather than at the Gauss integration points. A procedure was implemented in the FE program JAM to extrapolate and smooth Gauss point data to the element vertices, i.e.. corner nodes. Values at the midside nodes were then calculated from the values at the appropriate corner nodes.
The implemented smoothing procedure was developed and described by Hinton, Scon, and Ricketts (1975) and Hinton and Campbell (1974). The procedure is simple and straightforward. The smoothed stresses at the nodes may be calculated from the expression
5.1
51 "a b c b~ "I
d2
a3 • =
babe
c b a b * "u
*4 b c b a 0rv
where the a, are the smoothed stresses, at, au, am and ajy are the stresses
at the integration points, and a■ « 1 ♦ — , fc = - _ and c - 1 - -—
At a given corner node, smoothed values from adjacent elements are averaged to yield a single value.
Plane and Axisymmetric Verification Problems
Five problems with plane or axisymmetric geometries were solved with JAM to test and verify that the material models, the eight-node quadrilateral element, and numerous other algorithms were correctly implemented in the FE program. For each of the five problems, selected output from JAM are
64 Chapter 5 Natures and Verification of FE Program
compared with closed form or analytic solutions.
A Y
/I - nimm
A B c!Z ' rrnfmr ; 1
<- ->
Figure 5.1. Geometry and loading conditions for Verification Problem 1
Verification Problem 1 exercises a single element under distributed normal and shear loads of 1000/length as shown in Figure 5.1. The element sim- ulates an isotropic linear elastic material having a Young's modulus of 30 x 10 6 and a Poisson's ratio 0.3. The following boundary conditions were imposed:
ux * «y = 0 at point A and uy * 0 at points B and C
Tabl« 5.1. Results from Verification Problem 1
S tratt or
Strain
Plan« Strain Mana Straw
JAM Afntyfic JAM Anatytie
o. 1000. 1000. 1000 1000
o, 1000. 1000. 1000. •1000.
•« 1000. -1000 1000. 1000.
°t 600 600 0 0.
', • 1 733*10* • 1.733*10* -2.333*10* 2.333*10*
«V 1,733x10* 1 733*10* 2.333*10* 2.333*10*
^— -8.667*10* -6.667*10* •8.667*10* 8.667*10*
Table 5.1 compares stress and strain states calculated from the FE and
Chapter S Faaturas and Varttication of FE Program 65
X A
analytic solutions (Hibbitt, Karlsson and Sorensen 1989) for this problem under plane strain and plane stress boundary conditions. This problem exercises the elastic constitutive model, verifies that the eight-node quadratic element accurately models constant strain states, and also checks that distributed loads are correctly simulated. The results from JAM match the analytic solution exactly. Verification Problem 1 was also successfully solved using the Cap model algorithm.
L immii
1000
ABC
ttttttttt X 2 —>
m 7S
- V
Figure 5.2. Geometry and loading conditions for Verification Problem 2
Verification Problem 2 exercises a single axisymmetric element under distributed normal loads of 1000/area as shown in Figure 5.2. The element simulates an isotropic linear elastic material having a Young's modulus of 30 x 10 6 and a Poisson's ratio 0.3. The following boundary conditions were imposed:
u. = 0 at points A, B. and C
66
Table 5.2 compares stress and strain states calculated from the FE and analytic solutions (Hibbitt, Karlsson and Sorensen 1989) under the imposed axisymmetric boundary conditions. Like Verification Problem 1, problem 2 exercises the elastic constitutive model, verifies that the eight-node quadratic element accurately models constant strain states, and also checks that distributed loads are correctly simulated. The results from JAM match the analytic solution exactly.
Piane and axisymmetric patch tests were employed in Verification Problems 3 and 4. In the patch test, nodal point displacements are applied to a patch of elements such that a constant state of strain exists throughout the mesh. In Verification Problem 3, the elements simulate an isotropic linear elastic material having a Young's modulus of 30 x 10 6 and a Poisson's ratio 0.3. The imposed displacement boundary conditions were applied to the patch
Chapter 5 Features and Verification of FE Program
■a ■ *■-
Table 5.2. Results from Verification Problem 2
Strass or
Strain
Axisymmatric
JAM Analytic
°< -1000. -1000.
°i -1000. -1000.
°n 0. 0.
°S -1000. -1000.
*, -1.333 x10"5 -1.333 x10s
(i -1.333x10"5 -1.333 x10s
yn 0. 0.
<« -1.333 x 10s -1.333 x10"5
Y
.12
-V X K .24 > x
Figure 5.3. Geometry for Verification Problem 3
of elements shown in Figure 5.3 and were calculated as:
u = *> xlO and Ut x
y * - 2
xlO
Table S3 compares stress and strain states calculated from the FE and analytic solutions (Hibbitt. Karlsson and Sorensen 1989) for this problem under plane strain and plane stress boundary conditions. The results from JAM match the analytic solution exactly. Verification Problem 3 was also successfully solved using the Cap model algorithm.
Verification Problem 4 consists of a patch of axisymmetric elements
Chaotar 5 Faaturat and Vanficauon of Ft Program 67
Table 5.3. Results from Verification Problem 3
Strass or
Strain
Plana Strain Plana Strass
JAM Analytic JAM Analytic
°* 1600. 1600. 1333. 1333.
°v 1600. 1600. 1333. 1333.
"xv 400. 400. 400. 400.
at 800. 800. 0. 0.
fx 1.x 10'3 1.x 10'3 1.x 10-3 1.x 10"3
<v 1.x 10"3 1.x 10-3 1.x 10"3 1.x 10"3
1.x 10"3 1.x 10"3 1.x 10"3 1.x 10'3
Figure 5.4. Geometry for Verification Problem 4
simulating an isotropic linear elastic material having a Young's modulus of 30 x 10° and a Poisson's ratio 0.3. The imposed displacement boundary conditions were applied to the patch of elements shown in Figure 5.4 and were calculated as:
ur * Ur~ 1000) ♦£ 1 xlO 3 and u. - \z ♦ — 1000) xlO
Table 5.4 compares stress and strain states calculated from the FE and analytic solutions (Hibbitt. Karlsson and Sorensen 1989) for this problem. The results from JAM match the analytic solution exactly. Verification Problem 4 was also successfully solved using the Cap model algorithm.
68 Chapter 5 Faaturas and Verification of FE Program
Table 5.4. Results from Verification Problem 4
Stress or
Strain
Axisymmatric
JAM Analytic
°, 5.769 x10* 0.769x10*
°z 5.769x10* 5.769x10*
°n 1.154x10* 1.154x10*
°e 3.462x10* 3.462x10*
', 1.x 10'3 1.x103
ez 1.x Iff3 1. x 103
y<* 1.x Iff3 1.x 103
<» 0. 0.
I
• I
~igure 5.5. Mesh geometry for Verification Problem 5
Verification Problem 5 is a plane strain simulation of a thick wall cylinder subjected to an increasing internal pressure. Due to the symmetry of the problem, a quarter grid was used in the calculation; the problem geometry and FE mesh are shown in Figure S.S. The material was modeled with the following properties, a Young's modulus of 21000, a Poisson's ratio of 0.3, a yield stress of 24 and a linear hardening modulus of 0. When a cylinder with these properties is subjected to an increasing internal pressure above 10.4, an elastic-plastic boundary moves through the cylinder; on the external side of the boundary, all of the strains are elastic, and on the interior side, the strains are elastic-plastic. Table 5.5 compares stresses calculated from the FE and analytic solutions (Hodge and White 1950; Prager and Hodge 1951) at several radii within the elastic region for an applied internal pressure of 18, which places the elastic-plastic boundary at a radius of 160. The computed results
Chapter 5 Features and Verification of FE Program 69
• f
Table 5.5. Results from Verification Problem 5
Strass Radius
Plane Strain
JAM Analytic
Max. Principal Stress
or
°t
163.64 22.12 22.09
176.32 20.32 20.25
193.64 18.37 18.31
"i > 160 5.33 5.31
h > 160 23.11 23.03
are well within the imposed convergence tolerance of 1 percent. The results from JAM also agree with the results calculated by Owen and Hinton (1980) for this problem. This validates the plasticity formulation in JAM.
Consolidation Problems
To verify that the FE program could solve consolidation problems, output from JAM were compared to the results calculated from closed form solu-
/» n_
^*-5. 02 ^•v —~~?^_
1 o« ^* ~v»
1 >•. >^ I 06
• ra Solution \> A Sortoa Solution \ '
08 *\ *
00 02 04 0« 08 10 NormaNiod Por« Proaaur*
Figure 5.6. Pore pres sure versus depth at two time incremei us
tions. Boundary conditions and material properties were altered to fully exer- cise different features within the FE program. In Figure 5.6, depth versus calculated pore fluid pressures are plotted in a normalized format at two dif- ferent time increments for a one-dimensional consolidation problem in which
70 Chapter 5 Feature« and Verification of FE Program
the soil column was idealized as an elastic porous skeleton with an incompres- sible pore fluid. Good agreement is shown between the FE results and the closed form solution. A similar one-dimensional problem was solved with two materials having compressible pore fluids, where the ratios of pore fluid modulus to skeletal modulus (N) were 2000 (nearly incompressible pore fluid) and 5 (highly compressible pore fluid). Results from the FE program and the closed form solution (Chang and Duncan 1983) are plotted in Figure 5.7 as normalized displacements versus time factor, i.e., normalized time. Again,
1.0
-0.2
0.B -
0.6
S 0.4
£ 0.2 r-
0.0
10"
O FE Solution D FE Solution SartM Solution
N - 2000
10 -3 10 -2 10 -1 10° 101
Tim« Factor
Figure 5.7. Displacement versus time for one-dimensional consolidation of an idealized elastic material
the results show reasonable agreement between the FE results and the closed form solution.
A two-dimensional axisymmetric consolidation problem consisting of a circular foundation on a finite soil layer (Figure 5.8) was also calculated. The mesh is A units high by 10A units wide, and a uniform vertical load of radius A was applied to the top surfaces of three elements to simulate the foundation loads. The following boundary conditions were invoked for this problem. The vertical edges of the mesh (A-D and B-C) were constrained in the radial direction, the bottom edge of the mesh (C-D) was constrained in the vertical direction, the top surface (A-B) was free draining, and no flow conditions were applied to the three remaining surfaces (B-C. C-D. and A-D). The cal- culated settlements (in dimensionless format) from JAM (solid circles) are compared in Figure 5.9 to settlements calculated from the analytical solution
Chapter 5 Features and Verification of FE Program 71
A ■ B
C
Figure 5.8 Mesh geometry for Axisymmetric Consolidation Problem
0.26
0.» -
0.36
0.40 -
0.46
11 »III ill i i IIIIMI I i IIIIMI I i nun
0.60 | i MiiHq i inin^ i nniw| i inn
0.0001 0.001 0.01 0.1 1
Tim«
Figure 5.9 Displacement vs Time from 20 Consolidation Problem
(solid line). Again, there is excellent agreement between the calculation and the analytical solution.
72 Chapter S Features end Verification of FF. Program
Cryer Problem
A numerical simulation of Cryer's problem (Cryer 1963) was conducted as an additional verification test of the FE program. Cryer developed a closed
i
1 1 1 1 ' i i ,1 |
Rsdus
Figure 5.10. Mesh geometry for Cryer Problem
• •
Figure 5.11. Dimensionless pore pressure response for Cryer's problem
form solution to the problem of a sphere of elastic porous material loaded on the surface by a constant uniform pressure and having drained boundary con- ditions. For values of Poisson's ra .o less than 0.5, pore pressure at the center of the sphere increases to stress levels greater than the externally applied pressure and then dissipates. The greatest increase in pore pressure
Chapter 5 Feature« and Verification of FE Program 73
• A
occurs for values of Poisson's ratio equal to 0. This response is called the Mandel-Cryer effect after the two mathematicians who discovered the phe- nomena. Gibson et al. (1963) conducted laboratory experiments on clay spheres and were able to reproduce the Mandel-Cryer effect. They demon- strated that the total stress within a consolidating sphere is not time invariant as predicted by Terzaghi-Rendulic consolidation theory. Dimensionless total stress at the center of the sphere increases above unity and approaches unity at late time.
i
'? "• -
i
«>
/
ba^*'*»,,, ■
Figure 5.12. Dimensionless displacement response for Cryer's problem • <
Figure 5.13. Pore pressure contours for Cryer Problem
74 Chapttr 5 FMIUTM and Varrftcaiion of FE Program
• A
Utilizing the symmetry of the problem, the soil sphere was represented by the mesh depicted in Figure 5.10 and calculated as an elastic axisymmetric problem. A unit pressure was placed on the boundary during the first incre- ment of loading and held constant thereafter. Figure 5.11 and Figure 5.12 compare the results from JAM with the closed form solution for a value of Poisson's ratio equal to 0. Figure 5.11 is a plot of dimensionless pore pressure at the center of the sphere versus the square root of dimensionless time; Figure 5.12 is a plot of dimensionless displacement of th» outer surface of the sphere versus the square root of dimensionless time. The comparison between the FE calculation and the closed form solution is very good. Pore pressure contours at a dimensionless time of approximately 0.04 are plotted in Figure 5.13. At this early time, a significant portion of the sphere has pore pressures greater than unity.
Summary
In this chapter, the restart and post-processing features of the FE program were described. The documented verification problems indicate that the FE program correctly calculates one- and two-dimensional consolidation problems and elastic and elastic-plastic boundary value problems. Although the success- ful calculation of these verification problems does not certify the FE program is error free, they should increase the confidence of the end user.
Chapter 5 Features mv) Verification of FE Program 75
6 Numerical Simulations
76
L
Introduction
Numerical simulations of limestone behavior under drained and undrained boundary conditions are presented in this chapter. The ability of the 14-parameter cap model to simulate the basic drained behavior of limestone is demonstrated by comparing calculated responses of hydrostatic, uniaxial strain, and triaxial compression loadings with measured or recommended lime- stone responses. In a similar manner, the ability of the FE code to calculate the undrained behavior of limestone is demonstrated by comparing calculated responses of uniaxial strain loadings with recommended limestone responses. Finally, some example calculations are documented that demonstrate the utility of die FE code in analyzing laboratory test specimen conditions.
Salem Limestone
The limestone simulated in this chapter is commonly referred to as Salem, Bedford, or Indiana limestone. It was extracted from the Salem formation near the community of Bedford, Indiana. Mechanical property tests were conducted on intact specimens of 13.5-percent porosity Salem limestone by the Vermont office of Applied Research Associates. These mechanical property tests included drained and undrained (with pore pressure measurements) hydrostatic loading tests, K^ or uniaxial strain tests, triaxial compression tests, and strain path tests. Laboratory test data and recommended material proper- ties were obtained from Blouin and Chitty (1988a, 1988b) and Zelasko (1991).
Simulations
Process
Prior to numerically simulating limestone behavior under drained or undrained boundary conditions, the skeletal or drained behavior of Salem limestone was required. The 14-parameter cap model, which was documented in Chapter 3. was fit to recommended drained Salem limestone mechanical
Chapter 6 Num*»,c*i Stmuteuon*
properties. With this model and fit implemented into JAM, drained single- element boundary value problems were conducted to insure the FE code would reproduce the cap model calculations. Undrained single-element boundary value problems were then conducted.
Drained limestone simulations
The following recommended drained Salem limestone mechanical proper- ties were available for fitting: a failure envelope, hydrostatic load and unload behavior, K^ stress-strain behavior, K^ pressure-volume behavior and K^ stress paths, stress-strain curves from triaxial compression tests conducted at several confining stress levels, and strain path data along three different paths.
Typically, a high fidelity fit of both the hydrostatic loading and K„, or uniaxial strain, responses is impossible to capture with a relatively simple cap
Figure 6.1. Drained K0 stress-strain comparison
model. For this reason, greater emphasis was placed on fining the uniaxial- strain stress path and stress-strain responses and less emphasis on the hydro- static loading response. In Figures 6.1-6.3, the calculated drained K^ stress and strain responses from the 14-parameter cap model are compared to the recommended drained Kc behavior. Figure 6 1 compares the drained K„ stress-strain behavior. Figure 6.2 the K^ stress paths, and Figure 6.3 the K„ pressure-volume responses. The quality of the fits are very good. To make these fits, one must compromise between fining the KQ stress path and the K„ stress-strain behavior. The model K^ stress-strain response breaks over at a
Chtpw 6 Numeric«) Simulation* 77
■li ——-——-— .
5D0
ID
w UJ a If 3H0 "
t/l if)
a *•",
< a
„j
100 ?00 100 400 S00 bUP ~> 0 0
U&AN NORMAL STRESS, MEJa
Figure 6. 2. Drained K0 stress path comparison
. SOU
// /
3
": 13,:
^r A <&r il
if jr jff
? , "L/% ,'/ /
-■
3! ,' ^"^ M '
/ /
, i ' . i 4 ■ J
Figure 6.C J. Drained K0 pressure-volume comparison
vertical stress of approximately ISO MPa. A better match to the stress-strain behavior would require the Kg stress path to break over at a higher value of principal stress difference. Higher fidelity was desired in the K^ stress path.
78 Chapwt 6 Num»rir.ai Simulation*
riMMMM
In Figure 6.4, the calculated drained hydrostatic pressure-volume response of Salem limestone is compared to the recommended behavior. The quality of
500
400 7 •
a
300
•
a ■j->
5 a
?0 0
- " " s^ '
1
U;C
" / / " / ' / .•' * ^- / i i i
1 :< 3 d f
»öi'M'on MO» in osn-:fNT
Figure 6.4. Drained hydrostatic load-unload comparison
the fit is not very good because greater emphasis was placed on fitting the Kg behavior. Only a very complicated cap model, with several tens of fining parameters, would fit both the hydrostatic and KQ behavior of ihis material.
Drained triaxial compression (TXC) tests at confining pressures of 100 and 400 MPa were also simulated with the cap model. The calculated responses are plotted as principal stress difference versus axial strain and compared to actual test results in Figures 6.S and 6.6. The quality of the fits is quite good considering the error introduced into the calculations by the lack of fidelity in the calculated hydrostatic pressure-volt me response.
Undrained limestone simulations
The following single-element undrained simulations were performed using the Walker-Sternberg EOS for water and a carbonate EOS for the gram solids. The cap model fit to the recommended drained limestone properties modeled the skeletal behavior of the limestone.
An undrained KQ test conducted on a fully-saturated specimen of Salem limestone was simulated with the FE code. The output is compared to the available recommended properties is another method of verifying the FE code. The calculated and recommended stress-strain responses are compared
Chaptar 6 Numerical Simulation« 79
300
350
m
%
is a
fj
m
a
200
HO /'' '
> ' <
500
a
■so
] i J i ] i [ i 1 i
i
i
i ,j 1
( VFÜT:A, STOA.N HOC ft,'
TO
Figure 6.5. Drain ed TXC stress-strain comt »arison at 100 MPa confining
i
pressure
ft SI.
s
a
/ • / / / / / / / / / / / / / / / /
/ ' / ' / /
/ ' / / I '
- / i I 1 ft it
i 1 . 1 ' -»**•'■„.*. \ * h* S ►'UJN'
• •
Figure 6.6. Drained TXC stress-strain comparison at 400 MPa confining pressure
80
in a plot of total vertical stress versus total vertical strain (Figure 6.7). The calculated undrained K^ stress-strain response replicates the recommended
Chapter 6 Numeric») Sunultfiera
_i_ 3 9
0 /t
500 / /'
«00 / 7
/ 7
\ / If / 1
/ ' * 300
" 200
/ l / t / /
/ t s '
ft •
f / ............ u«ccxnmtn(i«d fit
too
(1 p c ? t o i ■ *i ;> c : '■ if 3 *■
*»M ■t»l Stf»i« *
Figure 6.7. Undrained K0 stress-strain comparison
• - «♦co***««a#c< *■
JO*; »SI-
*»( •*,-*# '(''IIS M- *
Figure 6.8. Undrained K0 stress path comparison
stress-strain response perfectly duhng die loading phase, while the unloading is slightly suffer. The calculated and recommended stress paths are compared in a plot of principal stress difference versus total mean normal stress
Chapttr 6 Hamm teal Simulation« 81
•
(Figure 6.8). The correlation between the calculated and the recommended stress paths is excellent.
The total (solid), effective (short dash) and pore fluid stresses (long dash) for the simulated undrained Kj, test are presented in Figure 6.9 in the format
:igure 6.9. Stresses in a simulated undrained K0 test
82
of stress versus total volume strain. This figure illustrates that, even in a competent rock such as limestone, a significant portion of the total applied stress is carried by the pore fluid, and the peak effective stress is only 20% of the peak total applied stress.
In "conventional" soil mechanics, water and the grain solids are often assumed to be incompressible. These assumptions have significant imflica- tions with regard to the response of materials during undrained loading. Under undrained or zero volume change boundary conditions, the undrained strength at failure and the undrained effective stress path are unique for a given material with prescribed initial conditions (Lambe and Whitman 1969). This means that the effective stress path is independent of the applied total stress path. A path of zero volume change in an elastic-plastic material implies that the elastic and plastic volume strains are of equal magnitude and opposite sign. To demonstrate tha» a unique effective stress path is developed, an undrained triaxial compression (TXC) tesi, i.e., constant radial stress durinc shear, and an undrained constant mean normal stress (CP) test, i.e., constant mean normal stress during shear, were numerically simulated. The following conditions existed prior to the undrained loading in both simula- tions: (1) the compressibilities of the water and the grain solids were zero;
Chapter 6 Numerical Simulations
JL ±
TXC TtSt CP Tut
ToUl-atraM path farCPtMt
Total-atnai path far TXC tart
100 150 JOO 250
Effect:!*« M»8fi Norm Str««i MPa
Figure 6.10. Total and effective stress paths from TXC and CP tests
(2) the nominal effective confining stress in the specimens was 200 MPa, which was generated by a drained hydrostatic loading; and (3) the initial pore fluid pressure was zero. The stresses during the shear loading were applied incrementally until the calculation would not converge under a convergence tolerance of 0.5 percent. The calculated total and effective stress paths for the TXC test (solid line) and the CP test (dashed line) are presented in Figure 6.10. The calculated effective stress paths from the TXC and CP tests are identical.
Additional undrained calculations were performed to prove that the un- drained effective stress paths are not unique when the water and grain solids are compressible. Three undrained TXC tests with the following initial condi- tions were simulated: the nominal effective confining stress was 200 MPa and the applied back pressures (initial pore fluid pressures) were 0, 100, and 300 MPa. The calculated effective stress paths are presented in Figure 6.11. These calculations indicate that as the applied back pressure increased from 0 to 300 MPa, the corresponding effective stress paths moved to lower values of effective mean normal stress. This response can be explained with the follow- ing logic. As the water becomes suffer with increasing levels of back pres- sure, equal strain increments within the specimen generate larger increments of pore fluid pressure. Thus, the effective stress paths move to the left in Figure 6.11.
The previous sections show that the FE code can accurately simulate both drained and undrained responses of Salem limestone under ideal laboratory test boundary condition. In the following sections, non-ideal boundary
Cftapta* 6 Numerical Simulation* 83
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JOO — — BP * JOG MPa
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A s v v\ - 100
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■ i i o ■" ice no ?oo ^o ion j'.c
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conditions that actually exist in the laboratory tests will be simulated.
Test Specimen Simulations • i
FEgrid
Cylindrical test specimens were simulated with the axisymmetric FE grid depicted in Figure 6.12. A quarter grid, consisting of 144 elements and 483 nodes, was used in the simulation due to the symmetric nature of the problem. The specimen end caps were included in the simulation to investigate the effects of end cap restraint upon both the stress and strain conditions within the test specimen. A worst case situation was simulated, i.e., one in which no sliding was permitted between the specimen and the steel end caps.
The simulated specimen is 11.43 cm (4.5 in.) in length with a diameter of 5.04 cm (2 in.). The permeability of the limestone was 1.03 x 10"9 m2, which is a value that insures a uniform pore pressure field throughout the mesh during both the drained and undrained simulations.
Simulation of drained triaxial compression teat
A drained triaxial compression test at a confining stress of 200 MPa was simulated in the following manner. First, equal increments of vertical and
84 Chapur 6 Numerical Simulations
J> ft
Axial Load/Displacement
1 End Cap
Specimen
/ WE Grid
End Cap
Confining Pressure
Figure 6.12. Finite element grid for specimen simulation
radial normal stresses were applied to the boundaries of the grid until the stresses equaled 200 MPa. Second, increments of vertical stress were applied until the total vertical stress reached 550 MPa. Finally, increments of vertical stress were removed until a hydrostatic state of total stress was obtained.
The output from this calculation is plotted in the form of contour plots of several stress or strain parameters, i.e., V^D« pl**tic volume strain, axial strain and radial strain (Figures 6.13 and 6.14). The contour plots present the state of stress or strain in the specimen at the time of peak total vertical stress (Note: the end cap is not included in these contour plots). With the exception of the upper 15 to 20
Table 6.1. Laboratory Calculated Strata and Strain Values
percent of the speci- men, i.e.. near the interface of the specimen and end cap, the state of stress within the specimen is relatively uniform (Figure 6 13) This is also true of the plastic volume strains within the specimen (Figure 6.13). However, both the axial and radial strains (Figure 6.14) exhibit significant gradients throughout the specimen. The smallest axial strains (0.03 m/m) are at the top of the specimen; the largest (0.20 m/m) develop at the center of the specimen. The
Anal Swam 15.8%
Radial Swam 6.2%
Principal Street Difference 310 MPa
VJ?o 179Mf»a
Chapter 0 Numerical Simulations 85
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radial strains vary from approximately zero at the specimen-end cap interface to as much as -0.065 m/m at the center of the specimen.
Table 6.1 contains values of stress and strain that would be calculated from laboratory measured load and deformation measurements. Stresses, e.g., principal stress difference, were corrected for the changing cross-sectional area of the test specimen. The stress values underestimate the strength of the test specimen by approximately 5 percent, 179 MPa (from above table) versus 190 MPa (average stress throughout specimen). The axial strain of 15.8 percent represents only a small portion of the calculated axial strain within the test specimen, while the radial strain of -6.2 percent is representative of the radial strains in the central portion of the test specimen.
This calculation implies that the state of stress within the test specimen is not significantly effected by end cap restraints. Uniform stresses occur throughout major portions of the specimen. In contrast, large axial and radial strain gradients were developed in the test specimen. This implies that some type of end-cap lubrication should be used if uniform states of strain are desired.
«D
Simulation of consolidated undrained triaxial compression test
A consolidated undrained triaxial compression test at a confining stress of 200 MPa was simulated with the FE code JAM. To begin the calculation, equal increments of vertical and radial normal stresses were applied to the boundaries of the grid until a hydrostatic stress of 200 MPa was achieved. During this hydrostatic loading, pore fluid was allowed to drain from the specimen. The boundary conditions were then changed so that no pore fluid could drain from the specimen. Finally, increments of vertical stress were applied until the solution would not converge, which implied that the specimen had failed. Failure occurred at a total axial strain of approximately 4.7 percent. A uniform pore fluid pressure existed throughout the specimen.
The output from this calculation is plotted in the form of contour plots of V)2D, volume strain, axial strain, and radial strain (Figures 6.15 and 6.16). The contour plots present the state of stress or strain in the specimen at the time of specimen failure (Note: The end cap is again not included in these contour plots). The Table 6.2. sfi2Q contours (Figure Laboratory Calculated Stress and Strain Values 6 15) illustrate that a for Undrained TXC Test uniform state of stress exists within a majority of the test specimen; significant gradients only exist near the specimen-end cap inter- face. The same is true of the volume strain
| Axial Strain 4-7% §
1 Radial Strain 0 4% 1
1 Volume Strain 3.8% §
1 Principal Stress DiHerence 216 MPa I
1 ' i L v J*0 124 MPa j
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contours. The calculated volume strains indicate that the specimen was compacting. The calculated pore pressures confirm this observation, i.e., they increase continuously until spec «en failure occurs. Due to the small axial strain level at which specimen failure occurs, significant gradients of axial and radial strain were not developed in the test specimen (Figure 6.16). Axial strains vary from 0.02 m/m at the top of the specimen to less than 0.0SS m/m in the center of the specimen. Radial strains range from approximately zero at the specimen-end cap interface to -0.004 m/m at the center of the specimen.
Table 6.2 contains values of stress and strain that would be calculated from laboratory measured load and deformation measurements. As in the drained simulation, stresses were corrected for the changing cross-sectional area of the test specimen. The laboratory calculated stress values correspond with the values from the test specimen simulation, i.e., 124 MPa (from above table) versus 123 MPa (average stress throughout specimen). The laboratory calcu- lated volume strain of 3.8 percent underestimates the simulated volume strains that vary between 4 and 4.4 percent throughout most of the test specimen. The axial strain of 4.7 percent agrees with the calculated axial strain throughout a major portion of the test specimen. The radial strain of -0.4% is representative of the simulated radial strains in the central portion of the test specimen.
This calculation demonstrates that significant stress and strain gradients are not developed in the limestone when the specimen fails at small axial strains, despite the introduction of end cap restraint. In addition, die stresses and strains calculated from laboratory measurements correlate well with the actual stress and strain states within the test specimen.
Summary
Numerical simulations of limestone behavior under drained and undrained boundary conditions were presented in this chapter. The ability of the 14-parameter cap model to simulate the drained behavior of the limestone was demonstrated by comparing calculated responses of hydrostatic, uniaxial strain, and triaxial compression loadings with measured or recommended limestone responses. The ability of the FE code to calculate the undrained behavior of the limestone was demonstrated by comparing calculated respon- ses of uniaxial strain loadings with recommended limestone responses. Finally, both drained and undrained TXC test simulations were documented which demonstrate the utility of the FE code in analyzing laboratory test specimen conditions.
Cham« 6 Numwical Smumom 91
7 Summary
This report documents the features and algorithms implemented into the FE code JAM. The FE code JAM is a numerical tool with the capability to:
• calculate strains, total and effective stresses, and pore fluid pressures for fully- and partially-saturated porous media,
• calculate the time dependent flow of pore fluids in porous media,
• model nonlinear irreversible stress-strain behavior, including coupled shear-induced volume change, and
• simulate the effect of nonlinear pore fluid compressibility and the contribution of the compressibility of the grain solids for stresses up to several hundred megapascals.
bi this report, the FE model implemented into JAM was described, and equations were developed for the residual forces. The features of the cap model and the relevant equations were documented, and the steps required to implement the cap model into the FE code were summarized. Other consti- tutive models available in the code were also reviewed.
The equations of state for air, water, and the grain solids were documented, and the equations for the compressibility of an air-water mixture were devel- oped. Several documented verification probten» demonstrated that die pro- gram works correctly. These problems included one- and two-dimensional consolidation problems, Cryer's problem of a consolidating sphere of soil, and a thick-walled cylinder problem.
Numerical simulations of limestone behavior under drained and undrained boundary conditions were presented. A 14-parameter cap model modelled the skeletal properties of the limestone Single element calculations demonstrated the ability of die FE code to simulate bom die drained and undrained re- sponses of the limestone under several different load and unload boundary conditions. The utility of the FE code was demonstrated by the simulation of drained and undrained triaxial compression tests, and the influence end cap restraint had on die stress and strain states in the test specimen.
92 Chapur 7 Summary
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4. TITLE AND SUBTITLE Two-Dimensional Finite Element Analysis of Porous Media at Multikilobar Stress Levels
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A finite element (FE) code was developed to verify laboratory test results or to predict unavailable laboratory test data for porous media loaded to multikilobar stress levels. The FE code simulates quasi-static, axisymmctric, laboratory mechanical property tests, i.e.. the laboratory tests are analyzed as boundan due problems. The code calculates strains, total and effective stresses, and pore fluid pressures for fully and partially saturated porous media. The time-dependent flow of the pore fluid is also calculated. An elastic-plastic strauvhardening cap model calculates the time-independent skeletal responses of the porous solids. This enables the code to model nonlinear irreversible stress-strain behavior and shear-induced volume changes. Fluid and solid compressibilities were incorporated into the code, and partially saturated materials were simulated with a "homogenized" compressible pore fluid. Solutions for several verification problems are given as proof that the program works correctly, and numerical simulations of limestone behavior under drained and undrained boundary conditions are also presented.
14 SUBJECT TERMS
Cap model Finite element
Compressibility Limestone
Effective streu Porous media
1? SECURITY CLASS* »CATION OF REPORT
UNCLASSIFIED
18 SECURITY CLASSIFICATION OF TH» PAGE UNCLASSIFIED
1» SECURITY CLASSIFICATION OF ABSTRACT UNCLASSIFIED
IS NUMBER OF PAGES 101
18 PRICE COOE
20 LaVKTATION OF ABSTRACT
I Farm 288 (ft*» 2-88) IM»»
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